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http://clang.llvm.org/docs/SanitizerSpecialCaseList.html
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# Sanitizer special case list¶
## Introduction¶
This document describes the way to disable or alter the behavior of sanitizer tools for certain source-level entities by providing a special file at compile-time.
## Goal and usage¶
User of sanitizer tools, such as AddressSanitizer, ThreadSanitizer or MemorySanitizer may want to disable or alter some checks for certain source-level entities to:
• speedup hot function, which is known to be correct;
• ignore a function that does some low-level magic (e.g. walks through the thread stack, bypassing the frame boundaries);
• ignore a known problem.
To achieve this, user may create a file listing the entities they want to ignore, and pass it to clang at compile-time using -fsanitize-blacklist flag. See Clang Compiler User’s Manual for details.
## Example¶
$cat foo.c #include <stdlib.h> void bad_foo() { int *a = (int*)malloc(40); a[10] = 1; } int main() { bad_foo(); }$ cat blacklist.txt
# Ignore reports from bad_foo function.
$clang -fsanitize=address foo.c ; ./a.out # AddressSanitizer prints an error report.$ clang -fsanitize=address -fsanitize-blacklist=blacklist.txt foo.c ; ./a.out
# No error report here.
## Format¶
Blacklists consist of entries, optionally grouped into sections. Empty lines and lines starting with “#” are ignored.
Section names are regular expressions written in square brackets that denote which sanitizer the following entries apply to. For example, [address] specifies AddressSanitizer while [cfi-vcall|cfi-icall] specifies Control Flow Integrity virtual and indirect call checking. Entries without a section will be placed under the [*] section applying to all enabled sanitizers.
Entries contain an entity type, followed by a colon and a regular expression, specifying the names of the entities, optionally followed by an equals sign and a tool-specific category, e.g. fun:*ExampleFunc=example_category. The meaning of * in regular expression for entity names is different - it is treated as in shell wildcarding. Two generic entity types are src and fun, which allow users to specify source files and functions, respectively. Some sanitizer tools may introduce custom entity types and categories - refer to tool-specific docs.
# Lines starting with # are ignored.
# Turn off checks for the source file (use absolute path or path relative
# to the current working directory):
src:/path/to/source/file.c
# Turn off checks for a particular functions (use mangled names):
fun:MyFooBar
fun:_Z8MyFooBarv
# Extended regular expressions are supported:
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2018-10-21 02:32:58
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https://quantumcomputing.stackexchange.com/questions/2267/how-to-implement-a-square-root-of-swap-gate-that-swaps-2n-qubits-on-the-ibm-q-c
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# How to implement a Square root of Swap gate that swaps 2n-qubits on the IBM Q (composer)?
I am looking for an implementation using the quantum gates provided by the IBM composer of the following quantum function:
• input $2n$ qubits
• output $2n$ qubits wherein $50\%$ of the cases the state of the $2$ sets of $n$ qubits are swapped and in the other $50\%$ of the cases the state of the $2n$ qubits remain unchanged. With swapped I mean that qubit $q[i]$ will get the state of qubit $q[n+i]$ and qubit $q[n+i]$ will get the state of $q[i]$. Note also that all the qubits must be swapped or not.
E.g. $n=3$: If input $|000111\rangle$ then output in 50% of the cases is $|000111\rangle$ and in the other 50% of the cases is $|111000\rangle$
We have already a solution when $n=1$ in the following StackOverflow question:
but how can we do that when $n>1$?
• I wonder if we can use a classical coin-flipper, and add an "if" statement in the QASM editor? – user1271772 Jun 8 '18 at 15:19
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2020-08-11 04:41:26
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https://hal.archives-ouvertes.fr/hal-01902850
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# The open LPC Paul trap for precision measurements in beta decay
Abstract : The LPCTrap experiment uses an open Paul trap which was built to enable precision measurements in the beta decay of radioactive ions. The initial goal was the precise measurement of the beta-neutrino angular correlation coefficient in the decay of$^{6}$He . Its geometry results from a careful optimization of the harmonic potential created by cylindrical electrodes. It supersedes previously considered geometries that presented a smaller detection solid angle to the beta particle and the recoiling ion. We describe here the methods which were used for the potential optimization, and we present the measured performances in terms of trapping time, cloud size and temperature, and space charge related limits. The properties of the ion cloud at equilibrium are investigated by a simple numerical simulation using hard sphere collisions, which additionally gives insights on the trapping loss mechanism. The interpretation for the observed trapping lifetimes is further corroborated by a model recently developed for ion clouds in Paul traps. The open trap shall serve other projects. It is currently used for commissioning purpose in the TRAPSENSOR experiment and is also considered in tests of the Standard Model involving the beta decay of polarized$^{23}$Mg and$^{39}$Ca ion in the frame of the MORA experiment. The latter tests require in-trap polarization of the ions and further optimization of the trapping and detection setup. Based on the results of the simulations and of their interpretation, different improvements of the trapping setup are discussed.
Document type :
Journal articles
Domain :
https://hal.archives-ouvertes.fr/hal-01902850
Contributor : Inspire Hep <>
Submitted on : Tuesday, October 23, 2018 - 10:09:37 PM
Last modification on : Wednesday, August 14, 2019 - 8:21:10 PM
### Citation
P. Delahaye, G. Ban, M. Benali, D. Durand, X. Fabian, et al.. The open LPC Paul trap for precision measurements in beta decay. Eur.Phys.J.A, 2019, 55 (6), pp.101. ⟨10.1140/epja/i2019-12777-3⟩. ⟨hal-01902850⟩
Record views
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2019-08-18 19:50:42
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https://discourse.cataclysmdda.org/t/batch-crafting-seems-broken/19618
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# Batch crafting seems broken
#1
Not that I’m complaining but something is off with batch crafting.
45 mins to do 14 batches of pemmican, should of taken 10hour 30 mins.
1 hour to do 3 short ropes that should of taken 4 hours and 30 mins
edit: 1 hour 30mins I guess since the log isn’t that precise after 1 hour.
#2
Try to update to the last experimental, I think a fix got pushed recently.
#3
Just noticed my bad, I haven’t saved yet so I’ll be able to fix it so I’m not getting free time too.
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2019-04-25 16:45:20
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https://mathoverflow.net/questions/107980/convolution-of-sequences
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# Convolution of sequences
Let for given real sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$, $c_n:=\sum_{k\in \mathbb Z} a_k b_{n-k}$ for $n \in \mathbb Z$ be the convolution of sequences $(a_n)$, $(b_n)$.
For classical convolution if one of two functions is in $L^p$, the second in $L^q$, where $1\leq p,q <\infty$ then their convolution $f*g$ is in $L^r$, where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$. Is it some similar type theorem true for convolution of sequences?
Yes. True not only for $\mathbb Z$ but for for abelian (more generally unimodular) locally compact group. (20.18) in Hewitt & Ross, Abstract Harmonic Analysis.
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2020-02-25 23:45:44
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https://fr.maplesoft.com/support/help/maplesim/view.aspx?path=NumberTheory%2FRadical
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Radical - Maple Help
# Online Help
###### All Products Maple MapleSim
NumberTheory
Radical
calculate the radical of an integer
Calling Sequence Radical( n )
Parameters
n - integer
Description
• The Radical( n ) function computes the radical of the nonzero integer n. The radical of a nonzero integer is the product of its prime divisors. The radical of any nonzero integer is, therefore, a square-free integer.
• The radical of $0$ is not defined; an exception is raised if 0 is passed.
• A positive integer is equal to its radical precisely when it is square-free.
Examples
> $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
> $\mathrm{Radical}\left(12\right)$
${6}$ (1)
> $\mathrm{Radical}\left(6\right)$
${6}$ (2)
> $\mathrm{Radical}\left(-12\right)$
${6}$ (3)
Compatibility
• The NumberTheory[Radical] command was introduced in Maple 2017.
• For more information on Maple 2017 changes, see Updates in Maple 2017.
See Also
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2023-03-26 03:13:26
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http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=8684&option_lang=eng
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RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
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Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find
Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 5, Pages 61–77 (Mi izv8684)
On an instantaneous blow-up of solutions of evolutionary problems on the half-line
M. O. Korpusov
Faculty of Physics, Lomonosov Moscow State University
Abstract: We consider some initial-boundary value problems on the half-line for ‘1+1’-dimensional equations of Sobolev type with homogeneous boundary conditions at the beginning of the half-line. We show that weak solutions of these problems are absent even locally in time. Moreover, we consider problems on an interval with the same boundary conditions on one of the ends of the interval $[0,L]$. We prove the local in time (unique) solubility of the problems under consideration in the classical sense, and obtain sufficient conditions for the blow-up of these solutions in finite time. Using the upper bounds thus obtained for the blow-up times for classical solutions of the corresponding problems, we show that the blow-up time tends to zero as $L\to+\infty$. Thus, a classical solution on the line is also absent, even locally, and we describe an algorithm for the subsequent numerical diagnosis of the instantaneous blow-up on the half-line.
Keywords: non-linear equations of Sobolev type, blow-up, local solubility, non-linear capacity, bounds for the blow-up time.
DOI: https://doi.org/10.4213/im8684
Full text: PDF file (612 kB)
First page: PDF file
References: PDF file HTML file
English version:
Izvestiya: Mathematics, 2018, 82:5, 914–930
Bibliographic databases:
UDC: 517.538
MSC: Primary 35L53; Secondary 35A01, 35B44
Citation: M. O. Korpusov, “On an instantaneous blow-up of solutions of evolutionary problems on the half-line”, Izv. RAN. Ser. Mat., 82:5 (2018), 61–77; Izv. Math., 82:5 (2018), 914–930
Citation in format AMSBIB
\Bibitem{Kor18} \by M.~O.~Korpusov \paper On an instantaneous blow-up of solutions of evolutionary problems on the half-line \jour Izv. RAN. Ser. Mat. \yr 2018 \vol 82 \issue 5 \pages 61--77 \mathnet{http://mi.mathnet.ru/izv8684} \crossref{https://doi.org/10.4213/im8684} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2018IzMat..82..914K} \elib{http://elibrary.ru/item.asp?id=36448772} \transl \jour Izv. Math. \yr 2018 \vol 82 \issue 5 \pages 914--930 \crossref{https://doi.org/10.1070/IM8684} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000448948200003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85056425976}
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2019-11-13 20:59:34
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https://ruor.uottawa.ca/handle/10393/6590
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Design, manufacturing and testing of hoop wound tubular in-plane shear specimens.
Description
Title: Design, manufacturing and testing of hoop wound tubular in-plane shear specimens. Authors: Dickson, Thomas E. Date: 1993 Abstract: High performance fibre reinforced polymer composite materials have become popular replacements for many traditional materials. Unfortunately, some of the methods used to determine composite material properties have not developed as quickly. In particular, there are presently more than ten different test methods used to determine in-plane shear properties. Many of these methods are criticized for their inability to produce a pure shear state in the test region. Tubular in-plane shear specimens, however, are generally accepted as the only specimens which do (at least theoretically) produce a pure shear state. In this study, design, manufacturing and testing procedures were developed for tubular in-plane shear specimens. A specimen was designed which included a thin-walled composite tube and an internal tapered bonded end fitting. In the manufacturing study, techniques were developed to fabricate a tube of excellent quality with high geometrical tolerances. Tubular in-plane shear specimens were then tested and the results compared to test results for three of the most popular tests in use today, namely, the $90\sp\circ$ Iosipescu, $10\sp\circ$ off-axis and $\pm$45$\sp\circ$ tensile in-plane shear test methods. The four tests were compared on the basis of their respective shear states and on their ability to produce shear modulus values which could be used to predict the tensile modulus of shear-sensitive laminates. Results indicate that, though the tubular shear specimens produce the most consistent state of pure shear, they do not accurately predict the tensile moduli of actual laminates. Of the five laminates tested, the $\pm$45$\sp\circ$ tensile shear test best predicted the tensile modulus of four laminates and was next best for the remaining laminate. It was found that as laminate shear sensitivity decreased, the suitability of different test methods changed. URL: http://hdl.handle.net/10393/6590http://dx.doi.org/10.20381/ruor-11349 Collection Thèses, 1910 - 2010 // Theses, 1910 - 2010
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2020-11-27 19:05:10
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https://www.rdocumentation.org/packages/frbs/versions/3.2-0/topics/HGD.update
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frbs (version 3.2-0)
HGD.update: FS.HGD updating function
Description
The role of this function is to update parameters within the simplified TSK fuzzy rule generation method using heuristics and the gradient descent method (FS.HGD). This function is called by the main function of the FS.HGD method, see FS.HGD.
Usage
HGD.update(data.train, miu.rule, func.tsk, varinp.mf, step.size = 0.01,
def)
Arguments
data.train
a matrix ($$m \times n$$) of normalized data for the training process, where $$m$$ is the number of instances and $$n$$ is the number of variables; the last column is the output variable.
miu.rule
a matrix with the degrees of rules which is the result of the inference.
func.tsk
a matrix of parameters of the function on the consequent part using the Takagi Sugeno Kang model. See rulebase.
varinp.mf
a matrix of parameters of membership functions of the input variables.
step.size
a real number between 0 and 1 representing the step size of the gradient descent.
def
a matrix which is obtained by the defuzzifier.
See Also
FS.HGD
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2021-04-11 07:42:06
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https://read.dukeupress.edu/demography/article/54/4/1305/167732/Fertility-Intentions-and-Residential-Relocations
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Abstract
This research addresses the question of whether fertility intentions (before conception) are associated with residential relocations and the distance of the relocation. We empirically tested this using data from two birth cohorts (aged 24–28 and 34–38 in the first survey wave) of the German Family Panel (pairfam) and event history analysis. Bivariate analyses showed that coupled individuals relocated at a higher rate if they intended to have a(nother) child. We found substantial heterogeneity according to individuals’ age and parental status, particularly for outside-town relocations. Childless individuals of average age at family formation—a highly mobile group—relocated at a lower rate if they intended to have a child. In contrast, older individuals who already had children—the least-mobile group—relocated at a higher rate if they intended to have another child. Multivariate analyses show that these associations are largely due to adjustments in housing and other living conditions. Our results suggest that anticipatory relocations (before conception) to adapt to growing household size are importantly nuanced by the opportunities and rationales of couples to adjust their living conditions over the life course. Our research contributes to the understanding of residential mobility as a by-product of fertility decisions and, more broadly, evidences that intentions matter and need to be considered in the analysis of family life courses.
Introduction
Growing evidence based on longitudinal data shows that the timing of residential relocations and childbearing are strongly interrelated. An explanation for this association suggests that if homes and neighborhoods are not perceived as adequate for childrearing, couples will adjust either their residential situation or their family plans within the given financial resources of the household. This so-called adjustment perspective conceives relocations occurring around the time of childbearing events as anticipations to changes in family size or as adaptations if conception has already occurred. However, relying on evidence of the timing of event occurrences sheds only partial light on the associations between fertility and spatial mobility. Although anticipatory moves may occur well in advance of child conception, fertility-induced relocation adjustments may also be delayed and occur long after the birth of the child when the household can finally afford the move, at about the time of a subsequent conception (Clark and Davies Withers 2009; Michielin et al. 2008). In addition, assumptions of research about the nature of the associations are deemed troublesome given the scant explicit evidence on the decision-making processes or the relations between relocations and conceptions beyond the correlation between the timing of events (Courgeau 1990; Kulu and Steele 2013).
Building on the existing body of research and following a life course approach combined with a decision-making perspective (Huinink and Kohli 2014), we focus on the underresearched subjective aspects of the association to examine whether a household relocation might be an outcome of fertility planning, which eventually may condition childbearing. Complementing prior research, we shed more light on how relocations are related to fertility processes, in anticipating conceptions, by examining the association between fertility intentions and residential mobility. Significant associations can be related to the solutions that households find to their needs to adjust where (i.e., housing conditions and residential environment), with whom (i.e., partnership circumstances), or how to raise children (i.e., accessibility to workplace or ties to friends and family). Further, because of increasing costs of both raising children and adequate housing, it is challenging for many—particularly young—households to adjust conditions through relocation in anticipation of the time when families are formed (Mulder 2006). In this study, we address these nuances in relocation behavior as anticipations to fertility events by examining the intersections of the association between relocations and fertility intentions with relevant indicators of family life course situation, such as individual’s age and parental status.
For the empirical analysis, we use a longitudinal research design with rich data on intentions and realization of childbearing as well as residential mobility within towns and beyond, over short and long distances. Prior research has found relocation distance to be a relevant factor: fertility events are more likely to be observed around shorter- than longer-distance relocations because short-distance relocations do not affect other daily activity patterns (Mulder and Hooimeijer 1999). We further distinguish between moves within or outside town. A common finding is that family formation, or the prospect of it, pushes couples to move short distances, outside town, toward less-dense areas deemed appropriate to raise children (Kulu 2008). We conduct an analysis of event histories on monthly detailed fertility (conception) and residential relocation episodes from six waves of the German Family Panel (pairfam). Results of our study demonstrate that accounting for fertility decision-making or planning stages yields further evidence on the conditions under which relocations take place. Our results also highlight both the importance of residential relocations in preparation for family events and the nuances of fertility-related relocations in relation to the family life stage situation. More generally, our research innovates by studying the associations between behavioral intentions in one life domain (fertility) and actual behavior in another one (spatial mobility). We show that intentions matter and need to be addressed in studies on the intersections between family life courses and other life domains.
Background
Extant literature offers important theoretical and empirical results about the interconnectedness of childbearing and residential transitions (for a recent review, see Wagner and Mulder 2015). Earlier studies have focused on the influence of local settings and spatial mobility on fertility behavior as well as on the relevance of household and family changes as a main motivation for spatial mobility (Courgeau 1985; Hervitz 1985; Huinink and Wagner 1989; Rossi 1955). Using a life course approach and longitudinal data, the basic hypotheses postulated by the early studies have been reexamined and enriched by recent research that has considerably contributed to advancing our knowledge on the associations between residential relocations and childbirth events (e.g., Clark and Davies Withers 2007; De Jong and Roempke Graefe 2008; Kulu 2005; Kulu and Milewski 2007; Kulu and Washbrook 2014).
A wealth of studies have examined post-relocation fertility and found that childbirth rates increase significantly after short-distance relocations and, to a lesser extent, after long-distance relocations. Often, the rationale behind these associations is the adjustment of living conditions (e.g., housing conditions or workplace accessibility) as a prerequisite for childbearing (Clark and Onaka 1983; Michielin and Mulder 2008; Wagner and Mulder 2015). This aligns well with the well-established housing ladder concept, which traces the tendency to move up to larger, owner-occupied and higher-status housing over the life course and with increasing family size (Clark et al. 1984; Mulder 2013). Along these lines, research has found that the likelihood of a birth increases after moving into owner-occupancy (Michielin and Mulder 2008; Mulder and Wagner 2001), a single-family dwelling (Feijten and Mulder 2002; Kulu 2008; Kulu and Vikat 2008), and rural areas (Courgeau 1989; Kulu 2008; Lindgren 2003). Moves induced by changes in couple household arrangements are also relevant to fertility behavior. For instance, the wish to start a family is positively associated with partners who previously lived in separate households to move in together (Kulu 2008; Wagner and Mulder 2014). Studies have also considered associations of fertility and workplace accessibility. Although there are no differences in fertility intentions by length of commuting to the workplace (Huinink and Feldhaus 2012), women who commute over long distances are more likely to be childless and to delay fertility (Meil 2010; Rüger et al. 2012). Not all relocations, particularly those over longer distances, are directly related to fertility behavior. Yet, after a couple’s long-distance relocation, often motivated by the job career of one partner, short-distance moves can follow to further improve the living conditions for eventual childrearing (Clark and Davies Withers 2007). In other cases, residential relocations (over short and long distances) are aimed at improving proximity to the extended family and previous places of residence where informal support for raising children is available (Hedman 2013; Michielin et al. 2008).
Several studies have also highlighted an immobilizing effect of families: the likelihood of a residential relocation out of town or across metropolitan areas generally decreases after the birth of a child, and even more so with increasing family size, because families may want to stay close to relatives and friends (Clark 2013; Kulu 2008; Wagner 1989). Additionally, the high relocation rates toward urban areas among young adults sink after childbirth because housing conditions and residential environment are deemed less adequate for families with young children (Courgeau 1990; Kulu 2008; Kulu and Boyle 2009; Kulu et al. 2009; Lindgren 2003). In contrast, moves within cities have been found to increase right after childbirth because families adjust dwelling size to family size or enter owner-occupancy if they have not already done so in anticipation (Clark and Huang 2003; Clark and Onaka 1983; Clark et al. 1994; Courgeau 1985).
Concurrent progresses in family life courses and housing ladders are often contested, though. Some research has highlighted tensions in the anticipated or adaptive home adjustments to childbirth. A tight housing market combined with insufficient financial resources constrains the ability of households to adapt their housing situation to a growing family (Clark and Huang 2003; Lersch 2014), perhaps explaining why many households relocate from expensive urban areas to more affordable suburbs or rural areas when the family grows (Kulu 2008). Research evidence has shown a negative association between house prices and fertility (Malmberg 2012). More generally, increasing costs of both raising children and housing have generated an endemic resource conflict that prevents many households from achieving their desired housing conditions, particularly homeownership, at the time they would like to have (more) children (Mulder 2006). Often, such conflict results in delayed fertility because housing security and housing stability remain important prerequisites for family formation (Kulu and Milewski 2007; Vignoli et al. 2013).
Evidence on substantive effects of the housing and residential conditions on fertility behavior suggests that both directions of causality deserve attention. Using a Finnish Longitudinal Fertility Register, Kulu and Steele (2013) applied multiprocess, multilevel event history models that controlled for unobserved factors commonly affecting both fertility and housing transitions. Findings of their analysis showed that childbirths increase the likelihood of home relocations, and relocations to single-family houses lead to higher rates of childbirth. Additionally, these associations were partly determined by unobserved factors, which call for caution when interpreting findings of previous analyses in light of potential selectivity.
In general, previous research has shown that residential mobility and fertility (behavior) are strongly interrelated. Many studies have focused on the realization of childbirth or pregnancy. Analytical attention has primarily been paid to adjustment mechanisms by which moves occurring around childbirth are meant to improve housing and spatial conditions. However, testing the adjustment mechanism based on the examination of the timing of events is contested because fertility-induced relocations may occur well in advance or much later than the observed childbirth. Courgeau and Lelièvre (1988) suggested using fuzzy time approaches to address interdependence across life domains. Along these lines, Clark and Davies Withers (2009) used time windows before and after childbirth and migration in order to assess anticipatory relocations caused by forthcoming childbirth and relocations that occur afterward. However, Courgeau (1990) acknowledged deficits of this approach to identify the causal logic of the associations and advocated the use of complementary information on the decision-making process. Following Courgeau’s suggestion, in this research we look into the possible associations between fertility intentions (the prior step in the family decision-making process) and residential mobility.
Theoretical Approach and Hypotheses
Our theoretical framework is based on the life course approach, supplemented with a fertility decision-making model (Huinink and Kohli 2014). From a life course perspective, individual life paths can be perceived as a sequence of individual actions and biographical transitions in different yet interdependent life domains. Aiming to generate or maintain subjective well-being as efficiently as possible, individuals plan and decide on activities in these life domains based on a subjective evaluation of their living conditions as well as goal-related aspirations and expectations. Gainful employment, raising children according to one’s standards, or owning a house are typical and interrelated lifetime goals that contribute to the improvement of subjective well-being.
We focus on fertility as a goal-seeking behavior over the life course. Successful parenthood is considered an ultimate goal by evolutionary scholars (Mace 2014). It can also be considered an intermediate goal to improve social well-being through affection, stimulation, and social approval arising from close parent-child relationships (Huinink and Kohli 2014). In contrast, spatial mobility is more of an instrumental goal (i.e., a tool) for generating well-being in other life domains (Huinink et al. 2014; Kley 2011; Willekens 1991). Residential relocations are often motivated by (expected) changes in employment or family size and, more generally, by improvements in quality of life (Geist and McManus 2008).
Considering not only respective events and activities but also the decision-making process preceding them, the question arises whether goal-related intentions in one life domain affect behavior (and intentions) in another life domain. This idea is pivotal in recent studies that combine the life course approach with decision-making theories to improve the understanding of interconnected life domains. Huinink et al. (2014) used an enriched subjective expected utility model (SEU) to examine the effects of intended and manifest behavior between spatial mobility and job mobility. They found that the interdependence across life domains extended beyond associations among observed events and also comprised the association between spatial mobility intentions and eventual job change.
Similarly, in this study, we shed light on preferences, resources, and subjective evaluations of available options with regard to family and place of residence. We postulate that residential mobility is instrumental to goal-seeking behavior such as fertility because it allows the adjustment to appropriate conditions for childrearing. When a pregnancy for a(nother) child is noticed, what we call an adaptive relocation (either before or after the birth) is likely to occur in order to adjust conditions (see Fig. 1). However, the intention of having a(nother) child (before conception) already leads to the consideration of where, how, and with whom to raise children, and—given that housing attributes, residential environment, and daily activity accessibility (e.g., distance to workplace or family) are perceived to be inappropriate—to an anticipatory relocation of the household.
Against this backdrop, fertility intentions could translate into relocations, particularly over short distances or within town, as a response to an anticipated change in the demand for living conditions (e.g., homeownership, dwelling type/size, and residential area) appropriate for raising children. The motivation to marry and/or move together (when living-apart-together) in order to enjoy parenting in a common household might also underlie an association between fertility intentions and relocation of one or both partners. Additionally, the need to reduce commute times in order to gain time resources for childcare or move to areas near relatives or childcare institutions could also support a significant association between fertility intentions and relocations, particularly over long distances. Adjustments in living conditions around the time of childbirth are well documented in the literature, conceptually and empirically, but the rationale of anticipatory relocations (when fertility is planned, before conception) as opposed to adaptive relocations (occurring during pregnancy or after childbirth) has not been clearly articulated. We argue that an anticipatory relocation is advantageous in that the “nest” is already well prepared before the demanding phase, in terms of both monetary and time resources, around the birth of the (next) child. Thus, a step-by-step strategy, whereby housing and location conditions are adjusted in advance of a(nother) child, supports couples by reducing stress due to the accumulation of partially competing tasks. Referring to this argumentation, our general hypothesis is that the intention to have a(nother) child has a positive association with residential relocations (fertility intentions hypothesis).
The associations between fertility intentions and relocations can be nuanced by combining the salience of the adjustments (i.e., how important and urgent are adjustments) and the availability of (economic) resources for relocation. For those intending to have a(nother) child, the salience of adjusting location is a function of the standards and adequacy of their current living conditions for childrearing. Despite the salience of an adjustment, relocations are expensive and require the availability of a certain level of resources. Couples with limited resources may have to postpone the relocation until either adjustments become more urgent or sufficient resources are accumulated. Thus, the fertility-related relocation may be delayed until conception occurs (adaptive relocation) or later (e.g., in anticipation of a subsequent child). Alternatively, the realization of fertility plans can be postponed until more resources have been accumulated.
Together with parental status, individuals’ age at (first) birth is a relevant life-stage factor moderating the relationship between family events and residential relocations (e.g., Geist and McManus 2008; Vidal and Lutz 2014). Shifts in contemporary fertility patterns due to extended periods of education, female labor force participation, and labor market insecurity have led couples to start families at increasingly older ages, after partners have consolidated their careers and have accumulated sufficient resources to start a family (Lesthaeghe 2010). Additional economic uncertainties partly amplified by the recent global financial crisis have translated into a stronger positive correlation between age at first birth and the socioeconomic situation of households (Kreyenfeld et al. 2012). Following this, recent evidence suggests that families experience increased uncertainty, economic hardship, and disadvantage, particularly young families (Buhr and Huinink 2012; Carlson 2012). Young people spend longer periods in temporary living arrangements and take more time to move to homeownership or good-quality housing (Lennartz et al. 2016).
The arguments about the salience and disposable resources across age groups and parental status lead us to posit another hypothesis about fertility-related anticipatory relocations: There is no general positive association between fertility intentions and residential relocations, but the association varies along with age and parental status of the individual (family life stage hypothesis).
Method
Data
We use data from six waves of the German Family Panel (pairfam, 2008–2013). This is an interdisciplinary, longitudinal study of intimate relationships and family living arrangements in Germany collecting data on the same respondents every year as well as additional information from their partners, parents, and children (Huinink et al. 2011). The original sample of pairfam comprises 12,402 persons and is representative of three birth cohort groups (1971–1973, 1981–1983, and 1991–1993) in 2008.
The data set is well suited for our research because it contains rich information on the timing, frequency, and distance of survey respondents’ residential relocations in each wave, which allows analyzing the process of residential mobility. In addition, information on partnership and family dynamics, including partners’ living arrangements, the intention to have a child, and information regarding fecundity, pregnancy, childbearing, and childbirth are available and collected over time. To our knowledge, no large, nationally representative household panel survey contains the level of detail on residential and fertility histories as well as fertility decision processes as this study. Response rates and attrition levels in pairfam are similar to those of other large scientific surveys run in Germany (Brüderl et al. 2015).1
Sample
Our analytical sample comprises respondents in a stable heterosexual couple, either living together with the partner or living-apart-together, who are able to have children but did not have children with any previous partner, who participated in the survey beyond the first wave, and who have complete information on key variables for the analyses.2 We also restrict the sample to the two older cohorts of the study, aged 24–28 and 34–38 years at the time of the first interview in 2008–2009. The median age at first childbirth in Germany was approximately 30 years for women and approximately 34 years for men of the birth cohort 1971–1973. Additionally, we find that most respondents of the youngest birth cohort (aged 15–17 in 2008–2009) did not intend to have children in the near future, which we attribute to dominant age norms on childbearing and the prioritization of education and career at this age. After exclusions, our sample consists of 1,442 respondents of the younger cohort (1981–1983), and 1,465 respondents of the older cohort (1971–1973).3 We find differences across birth cohorts in parental status and number of children in Wave 1. Less than 20 % of individuals of the younger cohort had children in the first wave, and more than 50 % of individuals of the older cohort already had children by then. However, the majority of individuals with children of both cohorts had only one child in Wave 1 (i.e., 75 % and 69 % of parents from the younger and older cohorts, respectively).
Measures
We consider three types of residential relocation events: (1) within-town relocations; (2) outside-town, short-distance relocations (if moves are less than 50km); and (3) outside-town, long-distance relocations (if moves are equal to or more than 50km, as in, e.g., Limmer and Schneider 2008). The data on residential relocations between towns is extracted from a retrospective questionnaire that was administered from Wave 2 onward. Detailed information on the month of relocation, place of origin, and destination allows us to construct measures of respondents’ residential relocations in monthly time intervals (coded as 1 if relocation occurs; coded as 0 otherwise) as well as the duration of residence. Within-town relocations are asked between Waves 2 and 6, but no detailed information on the month of the relocation is available. Random imputation of the month of within-town relocation occurrence (between the previous and the current survey wave) was performed using a uniform distribution.
The pairfam survey question on fertility intentions is asked annually to respondents who are self-rated fertile. The question item reads, “Do you intend to become a mother or father (again) over the next two years?”4 We collapse the response categories “yes, definitely” and “yes, maybe” into a positive response of an indicator variable (coded as 1), and the response categories “no, rather not,” “no, certainly not,” and “I have not had any thoughts about it” into a negative response (coded as 0).5 The fertility intentions indicator is time-varying and switches in the month when data is collected if the respondent answers differently than in her/his latest survey participation, or in the month when a pregnancy is observed. Because the question refers to a specific, relatively short period (i.e., two years), we expect that eventual relocations of those who intend to have (more) children might also be strongly related to fertility decisions.6 To control for fertility decisions, we use an indicator of pregnancy status leading to childbirth (coded as 1 for monthly intervals starting nine months before childbirth until the month when childbirth occurs; coded as 0, otherwise). To empirically address the family life stage hypothesis, we use an indicator for parental status (coded as 1 if the respondent already has a child/children; coded as 0 if the respondent is childless) and another indicator for age group (coded as 1 if the respondent was born in 1971–1973; coded as 0 if the respondent was born in 1981–1983), which distinguishes respondents in their mid- to late 30s from those in their mid- to late 20s in the first wave of the study.
For the multivariate models, we use a number of measures of covariates known to be associated with fertility (intentions) and residential relocations. These include time-varying housing and residential conditions, such as housing density (ratio between the number of people living in the household and the number of rooms), whether the respondent is living in owner-occupancy (indicator coded as 1; otherwise, 0), and settlement structure (two indicators for urban core, or inner city, and sparsely populated or rural areas; reference category is surrounding metropolitan areas, or suburbs). We also include time-varying variables of accessibility conditions, such as commuting time to work (two indicators for 30–50 minutes one way, more than 59 minutes one way; reference category is less than 30 minutes one way/no commute), and distance to parents (indicator coded as 1 if one-way drive to mother’s or father’s place of residence is more than 59 minutes one way; coded as 0 otherwise). To control for characteristics of union and household formation, we include time-varying indicators of marital status (coded as 1 if married; coded as 0 otherwise), coresidence status (coded as 1 if living-apart-together; coded as 0 otherwise), and partnership order within study window (coded as 1 if in second- or higher-order partnership; coded as 0 otherwise). Resources and resource conflicts are accounted for in time-varying covariates, such as level of education (indicator coded as 1 for higher education; coded as 0 otherwise), employment status (two indicators for part-time work and nonemployment situations; reference category is full-time work), partner’s level of education (indicator coded as 1 for higher education; coded as 0 otherwise) and employment status (indicator coded as 1 for full-time or part-time work; coded as 0 otherwise); household income (log-transformed annual income weighted by household size using an OECD equivalence scale); and number of children (two or more children). We derive other control variables from the literature, including time-constant covariates such as respondent’s sex (indicator coded as 1 if female; coded as 0 if male) or ethnic background (indicator coded as 1 if at least one parent born abroad; coded as 0 otherwise). We also derive a time-varying variable for whether the respondent resides in East Germany (coded as 1 if applies; coded as 0 otherwise).7 Summary statistics for model covariates (also broken down by fertility intentions and pregnancy status) can be reviewed in Table S2 in Online Resource 1.
Statistical Model
We use discrete-time event history analysis (EHA) to examine durations of partnered respondents’ residence leading to (1) within-town, (2) short-distance, or (3) long-distance relocations. The method is well-suited to our research because it allows estimating the associations with triggering factors that can be constant, status changing, or time-dependent on the occurrence of an event of interest that is observed within monthly time intervals (Blossfeld and Rohwer 1995). The model can be written as
$loghijRhij0=αRt+βRxijRt+ui,$
1
where hR is the discrete-time transition rate to a residential relocation of R type (R = 1, 2, 3), referring to within-town, short-distance, and long-distance relocations respectively. The reference outcome category is no relocation. The discrete-time hazard is defined as a multinomial response Y (coded as 1 to 3 if a relocation of type R is observed, and coded as 0 if no relocation is observed) for each discrete-time interval t given that no prior event within residential episode j and respondent i has been observed. The term αR(t) captures the duration of residence of a partnered respondent with the same partner in months since the start of a residential episode. We specify a piecewise linear transformation of the duration function that estimates the slope effect of residence duration for the following time intervals since the last relocation: 0 to 12 months, 13 to 36 months, and 37 months onward. Models including the piecewise specification of the duration function performed better (i.e., model fit) than those including linear or polynomial specifications of the duration variable. Additionally, the piecewise linear specification revealed nuances in the duration patterns across relocation types. An indicator for relocation episodes higher than first order was also included to capture average differences across episodes. Additionally, we include time-constant and time-varying covariates that capture the anticipated demands for housing and other living conditions in fully specified models. In accordance with the adjustment perspective, we expect anticipatory relocations will reflect household adjustment needs, so adding these covariates in the models will capture (at least part of) the association between fertility intentions and residential relocations. Time-constant and time-varying covariates and their coefficients are specified in$βiRxijRt$. Last, ui is a random term capturing correlations with time-constant, respondent-specific unmeasured characteristics.
Sensitivity Tests
It has been suggested that the causal association could also work in the reverse direction (Myers 2010). For instance, subjective housing insecurity and feelings of stress associated with residential relocations are known to negatively affect women’s fertility plans (Kulu and Milewski 2007; Vignoli et al. 2013). To address the potential selectivity of fertility decisions across those who relocate, we followed a strategy similar to Kulu and Steele (2013) by jointly estimating event history models for time to relocation and for time to conception leading to childbirth. Results of the sensitivity tests show that fertility intentions and pregnancy status coefficients of models that do not account for selectivity remain unchanged after controlling for time-constant unobserved factors that affect the timing of fertility and relocations. However, other sources of selectivity, such as time-varying unobserved factors affecting the association, may still be at play, preventing us from making causal assessments about the associations under analysis. Further details on the analytical model and results of the sensitivity test can be found in Online Resource 1, section B.
Results
Bivariate Results
Table 1 displays occurrences and rates of within-town, short-distance, and long-distance relocations across stages of fertility decision-making: that is, no intentions (reference category), fertility intentions, and pregnancy. We present unadjusted monthly rates (in thousands) in Table 1. Consistent with the literature, individuals relocate more often within town than leaving town; when leaving town, individuals relocate more often over short distances than over long distances. Relocations within town or over short distances are most frequent during pregnancy, and these also occur more often among those who intend to have a(nother) child than among those who do not. Long-distance relocations, in contrast, are not significantly more frequent during a pregnancy, but they are marginally more frequent when intending to have a(nother) child than when not intending to have a(nother) child.
Table 1 also portrays relevant differences in relocation patterns by an individual’s age and parental status. The younger age group (birth cohort 1981–1983, aged 24–28 at Wave 1) displays higher overall relocation rates than the older age group (birth cohort 1971–1973, aged 34–38 at Wave 1) if no fertility intentions are reported. Interestingly, differences in the relocation rates among those who report no fertility intentions, fertility intentions, and pregnancies are more marked for the older age group (who are at a late childbearing stage) than for the younger age group. The intention to have the first child spurs long-distance relocations for respondents at late family formation age (older age group) but deters short-distance and long-distance relocations for respondents at young family formation age (younger age group). After a conception leading to the birth of the first child, short-distance relocations occur at a higher rate for both younger and older individuals. The intention to have another child spurs relocations over any distance among older respondents, but this is not the case for respondents of the younger age group. After a conception leading to the birth of another child, within-town and long-distance relocation rates are significantly higher for respondents of the older age group, and no significant association is to be found for respondents of the younger age group.
Overall, unadjusted rates show that relocations occur at a higher rate during both fertility intentions (anticipatory relocations) and pregnancy stages (adaptive relocations), but the associations are uneven across age and parental status.
Multivariate Results
Table 2 presents results for within-town, short-distance, and long-distance relocation discrete-time hazard rates, respectively, which are jointly modeled as competing risks. In Model 1 (our baseline model) of Table 2, we first address the association of relocations with fertility intentions and pregnancy status, adjusting only for respondent’s age and parental status as well as the baseline durations, the relocation episode indicator, and one random variance term commonly specified for the three outcomes. Results of Model 1 show that after controlling for respondent’s age and parental status, the associations between fertility intentions and within-town, short-distance, and long-distance relocations are not statistically significant. In contrast, we find that pregnancies are positively associated with short-distance relocations. Already having children (reference category is childless couples) is significantly and negatively associated with relocations outside town but is not significantly related to within-town relocations. Being a member of the older age group is negatively associated with all types of relocations, and the associations are highly significant.
The nonsignificant associations between fertility intentions and relocations in Model 1 of Table 2 contrast with the significant bivariate associations (Table 1), suggesting that anticipatory relocations (before conception) are a function of different rationales, needs, and resources at different life stages. Additional models (not shown) with fewer covariates (i.e., only fertility intentions, pregnancy status, baseline duration, relocation episode indicator, and the random variance term) resulted in significant coefficients for fertility intentions and pregnancy status, confirming that the inclusion of respondent’s age group and parental status yielded insignificant fertility intentions and pregnancy status coefficients in the models presented in Table 2.
Model 2 (our fully specified model) in Table 2 adds a number of covariates for living conditions that are known predictors of both relocations and fertility decisions to Model 1. In brief, results show that when changes in living conditions are allowed for, neither the direction nor the significance of the covariates indicating the association between fertility intentions and relocations change. Instead, the coefficients of pregnancy status and family couple turn positive and significant for within-town relocations, the negative coefficient of family couple turns nonsignificant for short-distance relocations, and the negative coefficients of couple with children and older age group turn nonsignificant for long-distance relocations.
All in all, adjusted results for living conditions featured in the model covariates indicate that couples with children and older respondents relocate less because most adjustments may have already been made. Fertility intentions still are not significantly related to relocations. Interestingly, pregnancy status is positively related with within-town and short-distance relocations, revealing that further factors are associated with adjustment relocations of shorter distance. Additionally, the result also may point at the higher urgency for adjustment after conception (adaptive relocations) than before, during the planning stage (anticipatory relocations).
We briefly comment on selected coefficients in Model 2 of Table 2 that rendered interesting results. As expected, homeownership is one major deterrent of residential relocations, particularly outside town. Housing density (i.e., household size divided by number of rooms) is only significantly associated with within-town relocations. Interestingly, those living in the inner city are less likely to relocate outside town, over short distances. We find that a number of aspects of accessibility, such as lengthy commutes or a long distance from parents, are associated with long-distance relocations. Respondents in a living-apart-together couple arrangement also relocate more often outside town, both over short and long distances. In line with results of the couple migration literature, the education and employment status of the respondent and his/her partner are relevant to predict relocations over long distances.
We further assess whether respondents’ age and parental status moderate the associations between fertility intentions and relocations. In Fig. 2, we present selected results of a model equivalent to the ones presented in Model 1 (baseline model) and Model 2 (fully specified model) of Table 2 with additional two-way interaction terms for fertility intentions and age group, fertility intentions and parental status, and age group and parental status, as well as a three-way interaction term for the three variables. Equivalent interaction terms have been modeled for pregnancies with age group and parental status. Figure 2 shows results for full interactions of those not intending to have a(nother) child, those intending to have a(nother) child, and those who are (or whose partner is) pregnant in any of four situations: (1) younger age group, childless; (2) older age group, childless; (3) younger age group with child(ren); and (4) older age group with child(ren). The dots represent the discrete hazard rate of each group for each relocation type, and the spikes around the dot are the 95 % confidence interval (reference category is younger age group, childless, and not intending to have a child). Results of the interaction terms’ coefficients can be consulted in Table S3 in the Online Resource 1.
Despite a few significant coefficients for the interaction terms in multivariate models (see Table S3 in Online Resource 1), a graphical representation of the associations presented in Fig. 2 shows some variation within and between groups in relocation rates among those not intending to have a(nother) child and those intending to or those who are pregnant. In the baseline model, which adjusts only for residence duration and episode, we observe several statistically significant differences. In the full model, which additionally adjusts for a number of home and living conditions, we observe differences that are at most minimally significant. This finding confirms that after controlling for key predictors of residential relocation and fertility, variations in relocation rates by fertility intentions or pregnancy status over the life course are less marked, which evidences that anticipatory relocations are due to adjustments in housing and other living conditions.
Additional findings are worth considering. First, younger respondents intending to have a first child display a lower likelihood of outside-town relocations in the baseline model. That is, the 95 % confidence interval of the coefficients for fertility intentions in Fig. 2 (baseline model) does not cross the horizontal solid line. The significant difference vanished in the full model. Interestingly, for this group, outside-town relocations over short distances displayed a higher likelihood during pregnancy, suggestive of couples moving to (more affordable) suburbs outside the city as a reaction to—and not in anticipation of—the first child’s conception. Coefficients of younger respondents with children show no significant differences in relocation rates among those reporting no fertility intentions, fertility intentions, and being pregnant with another child.
Second, older childless respondents intending to have a(nother) child display a higher likelihood of long-distance relocations in the baseline model. That is, the 95 % confidence intervals of the coefficients for fertility intentions do not cross the 95 % confidence intervals of the coefficients for no fertility intentions within the individual’s age and parental status group. Among older respondents with children, fertility intentions also spur short-distance relocations. Additionally, older respondents with children move more often within town or outside town over long distances if a pregnancy is reported. Statistically significant differences in relocation rates among those reporting no fertility intentions (reference), fertility intentions, and being pregnant for a(nother) child vanish in the full models. The only exception is among respondents in a reported pregnancy stage, where higher likelihood of within-town relocations is at least marginally significant (p < .10).
Third, the comparison of coefficients shows interesting differences in relocation likelihoods across respondents of different age and parental status. If no fertility intention is reported, the relocation rates of older individuals are in many cases significantly lower than those of younger individuals. That is, among no fertility intention coefficients, the 95 % confidence intervals for old age group are lower and do not cross the ones for young age groups. Interestingly, differences in relocation rates across age groups are smaller among those who report fertility intentions or pregnancies. The coefficients for long-distance relocations even reverse, where older individuals display some slightly higher (but mostly nonsignificant) rates than younger individuals when reporting fertility intentions or pregnancies. Significant differences found in baseline models are again largely weaker in the full models. It is worth noting that despite controlling for a number of relevant covariates, fertility intentions and pregnancies among the older respondents with children relate to (marginally) significant higher within-town or long-distance relocation likelihoods than those of younger, childless respondents not intending to have children, which is a typical highly mobile group. This result again suggests that fertility-related adjustments might be partly postponed (or more salient) at later times, when families grow.8
Conclusions
The interdependencies between spatial mobility and fertility are complex. In this research, we shed more light on this by examining the associations between fertility intentions and residential relocations within town, or leaving town over short and long distances. Conceptually, we argued that the interdependence across life domains extends beyond associations among observed events, and acknowledged the relevance of preceding decision-making stages. We posited that relocations as solutions that households find to their needs for adjustment of where, with whom, or how to raise children can already be performed before an intended conception (anticipatory adjustment) and not only after conception (adaptive adjustment). Relocations in anticipation to changes in family size are deemed advantageous in that they support couples by reducing stress associated with the demanding phase around the birth of a child. We also acknowledged that associations between fertility intentions and relocations could be nuanced by the salience of the adjustments (i.e., how important and urgent they are) and the availability of (economic) resources for relocation. We expected these nuances to be relevant across family life stages (as a function of respondents’ age and parental status). We used longitudinal data from the German Family Panel (pairfam) and event history models to test the associations.
Some key findings arise from our study. First, unadjusted results showed that respondents who reported an intention to have a(nother) child relocated at a higher rate than those who did not. However, these associations mask compositional (family life course) differences between groups. Second, we found important heterogeneity, and even reversals, in the association between fertility intentions and relocation when respondents’ age and parental status are considered. This is particularly true for outside-town relocations, over short or long distances. Third, these associations emerged partly because of the diverse needs to adjust housing and other living conditions to changes in family size among groups of respondents.
With regard to the previous literature, a number of studies have claimed that residential relocations are a means to adjust housing and other living conditions in anticipation of (as well as in adaptation to) increases in family size. Our findings support these claims, with some evidence for vanishing differences in the relocation propensities of those who intend to have a(nother) child and those who do not intend to after controlling for a number of housing and other living conditions in the multivariate models (full models). However, we did not find that relocations in anticipation to changes in family size extend to all situations as suggested in our general fertility intentions hypothesis. In fact, the unadjusted positive correlations between fertility intentions and relocations vanished after adjusting only for respondent’s compositions of age and parental status (baseline models; i.e., not adjusted for housing and living conditions). Instead, consistent with research that demonstrates how economic uncertainties among young adults inhibit concurrent progress in family and housing life courses, our results (baseline models) showed that fertility intentions matter differently for relocations at various life stages, as suggested by the family life stage hypothesis. On the one hand, childless individuals around average age at family formation—an overall highly mobile group—relocate less often outside town if they intend to have a child. On the other hand, older individuals, particularly those who already have children—the least mobile group—relocate more often outside town if they intend to have another child. In line with the adjustment perspective, these differences vanished after we controlled for housing and other living conditions (full models). Differences in relocation rates according to fertility intentions were already less relevant for within-town relocations before controlling for covariates other than respondent’s age and parental status.
Our research contributes to the literature in a number of ways. First, by addressing the association between fertility intentions and relocation behavior, we improve the understanding of interdependent developments across different domains of life—that is, family and residence. We postulated and empirically tested for the first time that fertility intentions can trigger residential mobility. Our results complement previous studies that have relied on fuzzy time approaches (i.e., observation of event occurrences within a time window) to assess the relationships between fertility and relocation events. Because fertility intentions matter for couples’ relocation behavior under certain conditions, future studies may need to consider that the impact of fertility on relocations starts before the conception of a child, when families are planned. We examined not only differences in relocations over short and long distances but also differences in relocations within and outside town. Our results contribute to the literature in finding that couples relocate more often within town than outside town when intending to have a(nother) child. Additionally, we were able to provide more accurate evidence by using longitudinal data with detailed information on partnership dynamics (including living-apart-together situations), fertility decision-making stages, and residential histories.
Despite its contributions, our research could be improved in a number of ways. In a sensitivity test, we established that our results remained unchanged after adjusting for time-constant unobserved heterogeneity that commonly affects fertility and residential mobility. Nevertheless, we cannot rule out that other sources of selection—for example, due to time-varying unobserved heterogeneity—bias our results. Additionally, data limitations restricted the analysis to information mostly from one individual (i.e., main survey respondent) in each couple, despite growing recognition that partner’s intentions and characteristics have to be taken into account in equal measure to understand household decisions (Balbo et al. 2013). To further assess the interdependence of fertility and residential mobility, information on intentions to relocate should be considered as well. Future work should look at differences by gender in the associations between fertility intentions and spatial mobility. An incipient but growing literature on residential mobility to start coresidence with the significant other has found marked gendered patterns (Brandén and Haandrikman 2013).
Acknowledgments
This research was partially supported by the Deutsche Forschungsgemeinschaft (Grant No.VI711/1-1) and by the Australian Research Council Centre of Excellence for Children and Families over the Life Course (Project No. CE140100027).
Notes
1
The response rate in Wave 1 was 37 %, which led to the realization of 12,402 interviews. Attrition in Wave 2 was 22 % of the original sample, but attrition rates fell thereafter. By Wave 4, the attrition rate was 11 %, and the rate sank below 10 % in Wave 6. Younger age groups, changes in union status/new unions, or living-apart-together situations are confirmed correlates of panel attrition in pairfam (Müller and Castiglioni 2015).
2
We exclude 634 individuals who stated they were infertile, 98 individuals in nonheterosexual unions, 880 individuals who had children with ex-partners by the start of the study, 1,752 individuals who participated only in the first survey wave, and 240 individuals without response to the fertility intentions item in Wave 1. We also exclude 900 individuals who left the study by Wave 3 because retrospective information on residence started to be collected in Wave 3. After transforming the data set into person-months, we exclude person-months in episodes of separation/divorce (but include living-apart-together situations). As with other surveys, line-item nonresponse restricts the number of observations we use in our analysis. From the remaining 150,618 person-month records, we exclude approximately 20 % due to line-item missing values. The investigation of missing data patterns concluded that exclusions were mostly due to lack of information on household income, room stress, and lack of partner information (such as education and employment status in some survey waves). A descriptive analysis of key variables showed that their distributions are not affected by missing data (see Table S1, Online Resource 1). We have avoided multiple imputation strategies known to behave poorly, particularly for unbalanced longitudinal categorical data (Allison 2002).
3
Because no oversampling was performed in pairfam and variables that predict the probability of attrition are included in the multivariate analysis, we refrain from using weighting schemes that may lead to incorrect multivariate estimates. Descriptive analyses are also not weighed and hence are deemed sample summaries rather than population estimates.
4
Translated from the original question in German: “Haben Sie vor, in den nächsten zwei Jahren (erneut) Mutter bzw. Vater zu werden?”
5
It is fair to assume that those who have not had any thoughts about becoming a father or mother have no fertility intention because a fertility intention should require having thought about having a(nother) child.
6
For respondents who are pregnant or whose partner is pregnant at the time of the interview, the question refers to a subsequent parity. Because we consider intentions and pregnancies as two mutually exclusive states of fertility decision-making in the analysis, we switch the fertility intentions indicator to 0 during the pregnancy stage, but we re-switch the indicator after the birth of the child.
7
For marital status, cohabitation, living-apart-together situations, individual’s labor market status, and educational level, the transition month is known. For other time-varying covariates where the exact month of change in not known, response values switch in the month when data is collected if the respondent answers differently than in her/his latest survey participation. If a response is missing in a given wave, the interwave observations are excluded (see earlier sample exclusions). Additionally, for time-varying covariates referring to type of dwelling, residence location, distance to parents, and commuting response, values change in the relocation month using information from the subsequent survey wave.
8
The nuanced associations between the fertility intentions and residential relocations may hide finer moderated associations with available resources or specific resource-intensive adjustments than the associations presented here with family life stages. We addressed this question in further analyses including interactions of fertility intentions and pregnancies with indicators of couples’ resources (i.e., income, education, employment) and housing conditions (i.e., homeownership), but new interactions were not statistically significant, and other results remained unchanged (results available on request).
References
Allison, P. D. (
2002
).
Missing data
.
Thousand Oaks, CA
:
Sage publications
.
Balbo, N., Billari, F. C., & Mills, M. (
2013
).
Fertility in advanced societies: A review of research
.
European Journal of Population
,
29
,
1
38
. 10.1007/s10680-012-9277-y.
Blossfeld, H.-P., & Rohwer, G. (
1995
).
Techniques of event history modeling: New approaches to causal analysis
.
Mahwah, NJ
:
Lawrence Erlbaum Associates
.
Brandén, M., & Haandrikman, K. (
2013
).
Who moves to whom? Gender differences in the distance moved to a shared residence
(Stockholm Research Reports in Demography 2013:19).
Stockholm, Sweden
:
Department of Sociology, Demography Unit, Stockholm University
.
Brüderl, J., Schmiedeberg, C., Castiglioni, L., Arránz Becker, O., Buhr, P., Fuß, D., . . . Schumann, N. (
2015
).
The German Family Panel: Study design and cumulated field report (Waves 1 to 6), Release 6.0
(Pairfam Technical Paper No. 01).
Bonn, Germany
:
German Research Foundation
.
Buhr, P., & Huinink, J. (
2012
).
Die bedeutung familienpolitischer maßnahmen für die entscheidung zum kind [The importance of family policy for fertility decisions]
.
Zeitschrift für Sozialreform
,
58
,
315
341
. 10.1515/zsr-2012-0304.
Carlson, M. J. (
2012
).
Understanding young fertility in the context of economic disadvantage
. In A. Booth, S. L. Brown, N. S. Landale, W. D. Manning, & S. M. McHale (Eds.),
Early adulthood in a family context
(pp.
221
227
).
New York, NY
:
Springer
.
Clark, W. A. V. (
2013
).
Life course events and residential change: Unpacking age effects on the probability of moving
.
Journal of Population Research
,
30
,
319
334
. 10.1007/s12546-013-9116-y.
Clark, W. A. V., & Davies Withers, S. (
2007
).
Family migration and mobility sequences in the United States: Spatial mobility in the context of the life course
.
Demographic Research
,
17
(
article 20
),
591
622
. doi:10.4054/DemRes.2007.17.20.
Clark, W. A. V., & Davies Withers, S. (
2009
).
Fertility, mobility and labour-force participation: A study of synchronicity
.
Population, Space and Place
,
15
,
305
321
. 10.1002/psp.555.
Clark, W. A. V., Deurloo, M. C., & Dieleman, F. M. (
1984
).
Housing consumption and residential mobility
.
Annals of the Association of American Geographers
,
74
,
29
43
.
Clark, W. A. V., Deurloo, M. C., & Dieleman, F. M. (
1994
).
Tenure changes in the context of micro-level family and macro-level economic shifts
.
Urban Studies
,
31
,
137
154
.
Clark, W. A. V., & Huang, Y. (
2003
).
The life course and residential mobility in British housing markets
.
Environment and Planning A
,
35
,
323
339
. 10.1068/a3542.
Clark, W. A. V., & Onaka, J. L. (
1983
).
Life cycle and housing adjustment as explanations of residential mobility
.
Urban Studies
,
20
,
47
57
. 10.1080/713703176.
Courgeau, D. (
1985
).
Interaction between spatial mobility, family and career life-cycle: A French survey
.
European Sociological Review
,
1
,
139
162
. 10.1093/oxfordjournals.esr.a036382.
Courgeau, D. (
1989
).
Family formation and urbanization
.
Population: An English Selection
,
1
,
123
146
.
Courgeau, D. (
1990
).
Migration, family, and career: A life course approach
. In P. B. Baltes, D. L. Featherman, & R. M. Lerner (Eds.),
Life-span development and behaviour
(Vol.
10
, pp.
219
255
).
New York, NY
:
Lawrence Erlbaum
.
Courgeau, D., & Lelièvre, E. (
1988
).
Estimation of transition rates in dynamic household models
. In N. Keilman, A. Kuijsten, & A. Vossen (Eds.),
Modelling household formation and dissolution
(pp.
160
176
).
Oxford, UK
:
Clarendon Press
.
De Jong, G. F., & Roempke Graefe, D. (
2008
).
Family life course transitions and the economic consequences of internal migration
.
Population, Space and Place
,
14
,
267
282
. 10.1002/psp.506.
Feijten, P., & Mulder, C. H. (
2002
).
The timing of household events and housing events in the Netherlands: A longitudinal perspective
.
Housing Studies
,
17
,
773
792
. 10.1080/0267303022000009808.
Geist, C., & McManus, P. A. (
2008
).
Geographical mobility over the life course: Motivations and implications
.
Population, Space and Place
,
14
,
283
303
. 10.1002/psp.508.
Hedman, L. (
2013
).
Moving near family? The influence of extended family on neighbourhood choice in an intra-urban context
.
Population, Space and Place
,
19
,
32
45
. 10.1002/psp.1703.
Hervitz, H. M. (
1985
).
Selectivity, adaptation, or disruption? A comparison of alternative hypotheses on the effects of migration on fertility: The case of Brazil
.
International Migration Review
,
19
,
293
317
. 10.2307/2545774.
Huinink, J., Brüderl, J., Nauck, B., Walper, S., Castiglioni, L., & Feldhaus, M. (
2011
).
Panel Analysis of Intimate Relationships and Family Dynamics (pairfam): Conceptual framework and design
.
Zeitschrift für Familienforschung
,
23
,
77
101
.
Huinink, J., & Feldhaus, M. (
2012
).
Fertility and commuting behaviour in Germany
.
Comparative Population Studies
,
37
,
491
516
.
Huinink, J., & Kohli, M. (
2014
).
A life-course approach to fertility
.
Demographic Research
,
30
(
article 45
),
1293
1326
. doi:10.4054/DemRes.2014.30.45.
Huinink, J., Vidal, S., & Kley, S. (
2014
).
Individuals’ openness to migrate and job mobility
.
Social Science Research
,
44
,
1
14
.
Huinink, J., & Wagner, M. (
1989
).
Regionale lebensbedingungen, migration und familienbildung [Regional living conditions, migration and family formation]
.
Kölner Zeitschrift für Soziologie und Sozialpsychologie
,
41
,
669
689
.
Kley, S. (
2011
).
Explaining the stages of migration within a life-course framework
.
European Sociological Review
,
27
,
469
486
. 10.1093/esr/jcq020.
Kreyenfeld, M., Andersson, G., & Pailhé, A. (
2012
).
Economic uncertainty and family dynamics in Europe: Introduction
.
Demographic Research
,
27
(
article 28
),
835
852
. doi:10.4054/DemRes.2012.27.28.
Kulu, H. (
2005
).
Migration and fertility: Competing hypotheses re-examined
.
European Journal of Population
,
21
,
51
87
. 10.1007/s10680-005-3581-8.
Kulu, H. (
2008
).
Fertility and spatial mobility in the life course: Evidence from Austria
.
Environment and Planning A
,
40
,
632
652
. 10.1068/a3914.
Kulu, H., & Boyle, P. J. (
2009
).
High fertility in city suburbs: Compositional or contextual effects?
European Journal of Population
,
25
,
157
174
. 10.1007/s10680-008-9163-9.
Kulu, H., Boyle, P. B., & Andersen, G. (
2009
).
High suburban fertility: Evidence from four Northern European countries
.
Demographic Research
,
21
(
article 31
),
915
944
. doi:10.4054/DemRes.2009.21.31
Kulu, H., & Milewski, N. (
2007
).
Family change and migration in the life course: An introduction
.
Demographic Research
,
17
(
article 19
),
567
590
. doi:10.4054/DemRes.2007.17.19.
Kulu, H., & Steele, F. (
2013
).
Interrelationships between childbearing and housing transitions in the family life course
.
Demography
,
50
,
1687
1714
. 10.1007/s13524-013-0216-2.
Kulu, H., & Vikat, A. (
2008
).
Fertility differences by housing type: An effect of housing conditions or of selective moves?
Demographic Research
,
17
(
article 26
),
775
802
. doi:10.4054/DemRes.2007.17.26
Kulu, H., & Washbrook, E. (
2014
).
Residential context, migration and fertility in a modern urban society
.
,
21
,
168
182
. 10.1016/j.alcr.2014.01.001.
Lennartz, C., Arundel, R., & Ronald, R. (
2016
).
Younger adults and homeownership in Europe through the global financial crisis
.
Population, Space and Place
,
22
,
823
835
. 10.1002/psp.1961.
Lersch, P. M. (
2014
).
Residential relocations and their consequences: Life course effects in England and Germany
.
:
Springer
.
Lesthaeghe, R. (
2010
).
The unfolding story of the second demographic transition
.
Population and Development Review
,
36
,
211
251
. 10.1111/j.1728-4457.2010.00328.x.
Limmer, R., & Schneider, N. F. (
2008
).
Studying job-related spatial mobility in Europe
. In N. F. Schneider, & G. Meil (Eds.),
Mobile living across Europe I
(pp.
13
46
).
:
Barbara Budrich Publishers
.
Lindgren, U. (
2003
).
Who is the counter-urban mover? Evidence from the Swedish urban system
.
International Journal of Population Geography
,
9
,
399
418
. 10.1002/ijpg.296.
Mace, R. (
2014
).
When not to have another baby: An evolutionary approach to low fertility
.
Demographic Research
,
30
(
article 37
),
1074
1096
. doi:10.4054/DemRes.2014.30.37.
Malmberg, B. (
2012
).
Fertility cycles, age structure and housing demand
.
Scottish Journal of Political Economy
,
59
,
467
482
. 10.1111/j.1467-9485.2012.00590.x.
Meil, G. (
2010
).
Geographic job mobility and parenthood decisions
.
Zeitschrift für Familienforschung
,
22
,
171
195
.
Michielin, F., & Mulder, C. H. (
2008
).
Family events and the residential mobility of couples
.
Environment and Planning A
,
40
,
2770
2790
. 10.1068/a39374.
Michielin, F., Mulder, C. H., & Zorlu, A. (
2008
).
Distance to parents and geographical mobility
.
Population, Space and Place
,
14
,
327
345
. 10.1002/psp.509.
Mulder, C. H. (
2006
).
Home-ownership and family formation
.
Journal of Housing and the Built Environment
,
21
,
281
298
. 10.1007/s10901-006-9050-9.
Mulder, C. H. (
2013
).
Family dynamics and housing: Conceptual issues and empirical findings
.
Demographic Research
,
29
(
article 14
),
355
378
. doi:10.4054/DemRes.2013.29.14.
Mulder, C. H., & Hooimeijer, P. (
1999
).
Residential relocations in the life course
. In L. J. G. Van Wissen & P. A. Dykstra (Eds.),
Population issues: An interdisciplinary focus
(pp.
159
186
).
The Hague, The Netherlands
:
Springer
.
Mulder, C. H., & Wagner, M. (
2001
).
The connection between family formation and first-time home ownership in the context of West Germany and the Netherlands
.
European Journal of Population
,
17
,
137
164
. 10.1023/A:1010706308868.
Müller, B., & Castiglioni, L. (
2015
).
Attrition im Beziehungs und Familienpanel pairfam
[Attrition in the German Family Panel pairfam]. In J. Schupp & C. Wolf (Eds.),
Nonresponse bias
(pp.
383
408
).
:
Springer VS
.
Myers, S. M. (
2010
).
Connecting the demographic dots: Geographic mobility and birth intentions
.
Journal of Family Issues
,
31
,
1622
1651
. 10.1177/0192513X10365302.
Rossi, P. H. (
1955
).
Why families move: A study in the social psychology of urban residential mobility
.
Glencoe, IL
:
Free Press
.
Rüger, H., Feldhaus, M., Becker, K. S., & Schlegel, M. (
2012
).
Circular job-related spatial mobility in Germany: Comparative analyses of two representative surveys on the forms, prevalence and relevance in the context of partnership and family development
.
Comparative Population Studies
,
36
,
193
220
.
Vidal, S., & Lutz, K. (
2014
,
June
).
Continuities and changes in internal migration biographies in West Germany: An analysis of event sequences
.
Paper presented at the European Population Conference
,
Budapest, Hungary
.
Vignoli, D., Rinesi, F., & Mussino, E. (
2013
).
A home to plan the first child? Fertility intentions and housing conditions in Italy
.
Population, Space and Place
,
19
,
60
71
. 10.1002/psp.1716.
Wagner, M. (
1989
).
Räumliche mobilität im lebensverlauf: Eine empirische untersuchung sozialer bedingungen der migration
[Residential mobility over the lifecourse: An empirical investigation of the social conditions of migration].
Stuttgart, Germany
:
Ferdinand Enke
.
Wagner, M., & Mulder, C. H. (
2014
,
June
).
Understanding the transition from living apart together to a cohabitation—Who moves to establish a co-residential partnership?
Paper presented at the European Population Conference
,
Budapest, Hungary
.
Wagner, M., & Mulder, C. H. (
2015
).
Spatial mobility, family dynamics, and housing transitions
.
Kölner Zeitschrift für Soziologie und Sozialpsychologie
,
67
,
111
135
. (in German)10.1007/s11577-015-0327-4.
Willekens, F. J. (
1991
).
Understanding the interdependence between parallel careers
. In J. J. Siegers, J. de Jong-Gierveld, & E. van Imhoff (Eds.),
Female labour market behaviour and fertility
(pp.
11
31
).
Berlin, Germany
:
Springer
.
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### Q: what book on Bayesian statistics, preferably with R? [duplicate]
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### How did you learn Bayesian statistics and what would you recommend as a reliable source? [duplicate]
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### Bayesian references [duplicate]
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### recommended books for preliminary concepts of Bayesian Statistics [duplicate]
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### What is a good book about the philosophy behind Bayesian thinking?
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### Checking whether accuracy improvement is significant
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### Use of prior and posterior predictive distributions?
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### Which expansions and identities are useful to applied statisticians? [closed]
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### “Bayesian and Frequentist Regression Methods” by Jon Wakefield, a good introductory Bayesian textbook for frequentist economics graduates?
Here's a link to a good question regarding Textbooks on Bayesian statistics from some time ago. People suggested John Kruschke's "Doing Bayesian Data Analysis: A Tutorial Introduction with R and BUGS"...
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### Composition of probability density
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### Bayesian analysis: Estimate whether a parameter is 0 or not
I have the following problem: I need to assess whether a given parameter $B$ is equal to 0. Let's consider the following model (my problem is more complicated but I think that this example is ...
My question is about how to actually do this both rigorously and practically. Allow me to elaborate. Suppose that we have data $(x_1,y_1),...,(x_N,y_N) \in \mathbb{R}^p \times \{0,...,k-1 \}$. I'd ...
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2020-01-27 06:05:06
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https://practicepaper.in/gate-ee/complex-variables
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# Complex Variables
Question 1
The real numbers, x and y with $y = 3x^2 + 3x + 1$, the maximum and minimum value of y for $x \in [-2, 0]$ are respectively ________
A 7 and 1/4 B 7 and 1 C -2 and -1/2 D 1 and 1/4
GATE EE 2020 Engineering Mathematics
Question 1 Explanation:
\begin{aligned} y&=3x^{2}+3x+1 \; \; in \; [-2,0] \\ \frac{\partial y}{\partial x}&=6x+3,\; \; \frac{\partial^2 y}{\partial x^2}=6 \\ \frac{\mathrm{d} y}{\mathrm{d} x}&=0\\ 6x+3&=0 \\ x&=\frac{-1}{2} \\ \frac{d^{2} y}{dx^{2}}&=6 \gt 0\text{ minimum} \end{aligned}
Maximum value of y in [-2, 0] is maximum {f(-2), f(0)}
max{7, 1} = 7
Minimum value of y in [-2, 0]
$min\begin{Bmatrix} f(-2) & f(0) &f(-\frac{1}{2}) \\ \downarrow, &\downarrow, &\downarrow \\ 7& 1 & \frac{1}{4} \end{Bmatrix}+=\frac{1}{4}$
Maximum value 7, minimum value $\frac{1}{4}$
Question 2
The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is:
$\int_{c}\frac{z^2+1}{z^2-2z}dz$
A $8 \pi i$ B $-8 \pi i$ C $- \pi i$ D $\pi i$
GATE EE 2020 Engineering Mathematics
Question 2 Explanation:
\begin{aligned} I&=\int _C \frac{z^2+1}{z^2-2z}dz\;\;\;|z|=1 \\ \text{Using } & \text{Cauchy's integral theorem}\\ \int _C\frac{F(z)}{z-a}dz&=2 \pi i (Re_{(z=a)})\;\;\;...(i)\\ I&=\int _C \frac{z^2+1}{z(z-2)}dz \end{aligned}
Poles are at z=0 and 2 but only z=0 lies inside the unit circle.
Residue at $(z=0)=\lim_{z \to 0}\frac{z^2+1}{z(z-2)}$
$Re_{(z=0)}=-\frac{1}{2}$
Using equation (i)
$\int _C \frac{z^2+1}{z^2-2z}dz=2 \pi i \times \left ( \frac{-1}{2} \right )=-\pi i$
Question 3
Which of the following is true for all possible non-zero choices of integers $m, n; m \neq n$, or all possible non-zero choices of real numbers $p, q ; p\neq q$, as applicable?
A $\frac{1}{\pi}\int_{0}^{\pi}\sin m\theta \sin n\theta \; d\theta =0$ B $\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\sin p\theta \sin q\theta \; d\theta =0$ C $\frac{1}{2\pi}\int_{-\pi}^{\pi}\sin p\theta \cos q\theta \; d\theta =0$ D $\lim_{\alpha \to \infty }\frac{1}{2\alpha }\int_{-\alpha }^{\alpha }\sin p\theta \sin q\theta \; d\theta =0$
GATE EE 2020 Engineering Mathematics
Question 3 Explanation:
\begin{aligned} \because \; p& \neq q\\ &\frac{1}{2\pi}\int_{-\pi}^{\pi} \sin p\theta \cos q\theta d\theta \\ &=\frac{1}{2\pi}\cdot \frac{1}{2}\int_{-\pi}^{\pi} [\sin (p+q)\theta + \sin (p-q)\theta] d\theta \\ &=\frac{1}{4\pi}\left [ \frac{-1}{(p+q)}\cos (p+q)\theta -\frac{1}{(p-q)}\cos (p-q)\theta \right ]_{-\pi}^{\pi}\\ &=\frac{-1}{4\pi} \left \{ \frac{1}{(p+q)}(\cos (p+q) \pi -\cos (p+q)(-\pi)) \right.\\ &+\left. \frac{1}{(p-q)}(\cos (p-q) \pi -\cos (p-q)(-\pi)) \right \}\\ &=0 \end{aligned}
Question 4
$ax^3+bx^2+cx+d$ is a polynomial on real $x$ over real coefficients $a, b, c, d$ wherein $a \neq 0$. Which of the following statements is true?
A d can be chosen to ensure that x = 0 is a root for any given set a, b, c. B No choice of coefficients can make all roots identical. C a, b, c, d can be chosen to ensure that all roots are complex. D c alone cannot ensure that all roots are real.
GATE EE 2020 Engineering Mathematics
Question 4 Explanation:
Given Polynomial $ax^{3}+bx^{2}+cx+d=0;\; \; \; a\neq 0$
Option (A):
If d=0, then the polynomial equation becomes
\begin{aligned} ax^3+bx^2+cx&=0\\ x(ax^2+bx+c)&=0 \\ x=0 \text{ or } ax^2+bx+c&=0 \end{aligned}
d can be choosen to ensure x=0 is a root of given polynomial.
Hence, Option (A) is correct.
Option B:
A third degree polynomial equation with all root equal is given by
$(x+\alpha )^3=0$
Thus, by selecting suitable values of a, b, c and d we can have all roots identical.
Hence, option (B) is incorrect.
Option (C): Complex roots always occurs in pairs,
So, the given polynomial will have maximum of 2 complex roots and 1 real root.
Hence, option (C) is incorrect.
Option (D): Nature or roots depends on other coefficients also apart from coefficient 'c'.
Hence, option (D) is correct.
Hence, the correct options are (A) and (D).
Question 5
The closed loop line integral
$\oint _{|z|=5}\frac{z^3+z^2+8}{z+2}dz$
evaluated counter-clockwise, is
A $+8j\pi$ B $-8j\pi$ C $-4j\pi$ D $+4j\pi$
GATE EE 2019 Engineering Mathematics
Question 5 Explanation:
\begin{aligned} \oint _{|z|=5}\frac{z^3+z^2+8}{2+2}dz&=2 \pi j \text{ (sum of residues)} \\ &=2 \pi j \times \left [ \lim_{z \to -2} \frac{(z+2)(z^3+z^2+8)}{(z+2)}\right ] \\ &= 2 \pi j \times \left [ \frac{-8+4+8}{1} \right ]=8 \pi j \end{aligned}
Question 6
Which one of the following functions is analytic in the region $|z|\leq 1$?
A $\frac{z^2-1}{z}$ B $\frac{z^2-1}{z+2}$ C $\frac{z^2-1}{z-0.5}$ D $\frac{z^2-1}{z+j0.5}$
GATE EE 2019 Engineering Mathematics
Question 6 Explanation:
By Cauchy integral theorem,
$\int \frac{z^2-1}{z+2}dz=0$
Therefore, $\frac{z^2-1}{z+2}$ is analytic in the region $|z|\leq 1$.
Question 7
If C is a circle |z|=4 and $f(z)=\frac{z^{2}}{(z^{2}-3z+2)^{2}}$, then $\oint_{C}f(z)dz$ is
A 1 B 0 C -1 D -2
GATE EE 2018 Engineering Mathematics
Question 7 Explanation:
\begin{aligned} &\int \frac{z^2}{(z^2-3z+2)^2}dz\\ &\int \frac{z^2}{(z-1)^2(z-2)^2}dz\\ \end{aligned}
\begin{aligned} \text{Res. }f(z)_{z=1}&=\lim_{z \to 1}\frac{1}{1!}\frac{d}{dz} \left ((z-1)^2\frac{z^2}{(z-1)^2(z-2)^2} \right )\\ &=\lim_{z \to 1}\left (\frac{2z(z-2)^2-2z^2(z-2)}{((z-2)^4} \right )\\ &=\lim_{z \to 1}\left (\frac{2z(z-2)-2z^2}{((z-2)^3} \right )=\frac{-4}{-1}=4\\ \text{Res. }f(z)_{z=2}&=\lim_{z \to 2}\frac{1}{1!}\frac{d}{dz} \left ((z-2)^2\frac{z^2}{(z-1)^2(z-2)^2} \right )\\ &=\lim_{z \to 2}\left (\frac{(z-1)^2\cdot 2z-z^22(z-1)}{((z-1)^4} \right )\\ &=\lim_{z \to 2}\left (\frac{2z(z-1)-2z^2}{((z-1)^3} \right )=\frac{4-8}{1}=-4\\ \end{aligned}
By residue theorem, $I=2 \pi i (4-4)=0$
Question 8
The value of the integral $\oint_{c} \frac{z+1}{z^{2}-4}dz$ in counter clockwise direction around a circle C of radius 1 with center at the point z=-2 is
A $\frac{\pi i}{2}$ B $2\pi i$ C $-\frac{\pi i}{2}$ D $-2\pi i$
GATE EE 2018 Engineering Mathematics
Question 8 Explanation:
\begin{aligned} &\int \frac{z+1}{z^2-4}dz\\ &\int \frac{z+1}{(z-2)(z+2)}dz\\ &\int \frac{\frac{z+1}{z-2}}{z+2}dz \;\; \text{where, } f(z)=\frac{z+1}{z-2}\\ &=2 \pi i f(-2)\\ &=2 \pi i \left ( \frac{-2+1}{-2-2} \right )\\ &= 2 \pi i \left ( \frac{-1}{-4} \right )=\frac{\pi i}{2} \end{aligned}
Question 9
The value of the contour integral in the complex-plane
$\oint \frac{z^{3}-2z+3}{z-2}dz$
Along the contour |Z|=3, taken counter-clockwise is
A $-18 \pi i$ B 0 C 14$\pi i$ D 48$\pi i$
GATE EE 2017-SET-2 Engineering Mathematics
Question 9 Explanation:
Pole, z=2 lies inside |z|=3
$\text{Res }f(z)=\lim_{z \to 2}(z-2)\frac{z^2-2z+3}{z-2}$
$z=2, \;\;\;\;\; =8-4+3=7$
By Cauche redisue theorem,
$I=2 \pi i (7)=14 \pi i$
Question 10
Consider the line integral $I=\int_{c}(x^{2}+iy^{2})dz$ where $z = x + iy$. The line C is shown in the figure below.
The value of I is
A $\frac{1}{2}i$ B $\frac{2}{3}i$ C $\frac{3}{4}i$ D $\frac{4}{5}i$
GATE EE 2017-SET-1 Engineering Mathematics
Question 10 Explanation:
From the diagram C is $y=x$
\begin{aligned} I &=\int _C (x^2+iy^2)dz\\ &= \int _C (x^2+iy^2)(dx+idy)\\ &= \int _C(x^2++ix^2)(dx+idy)\\ &= \int _Cx^2 \; dx+ix^2\; dx+ix^2 \; dx-x^2 \; dx\\ &= 2i\int_{0}^{1}x^2 \; dx=\left.\begin{matrix} 2i\left ( \frac{x^3}{3} \right ) \end{matrix}\right|_0^1=\frac{2i}{3} \end{aligned}
There are 10 questions to complete.
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2022-09-25 07:44:34
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https://gmw.globalmathproject.org/station/I5SO
|
05 530 383
### Question 1
With pencil and paper, use dots-and-boxes to compute $2130 \div 10$.
Can you explain why, with unexplosions, the answer will be $213$? Do look for groups of ten in your picture.
Most people say that to divide a number that ends with zero by ten, just cross off the final zero from the number. Can you explain now why that action is sure to lead to the correct answer?
### Let's Go Wild!
Register NOW and unlock all islands!
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2021-03-08 06:22:55
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https://deepai.org/publication/the-bayesian-method-of-tensor-networks
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# The Bayesian Method of Tensor Networks
Bayesian learning is a powerful learning framework which combines the external information of the data (background information) with the internal information (training data) in a logically consistent way in inference and prediction. By Bayes rule, the external information (prior distribution) and the internal information (training data likelihood) are combined coherently, and the posterior distribution and the posterior predictive (marginal) distribution obtained by Bayes rule summarize the total information needed in the inference and prediction, respectively. In this paper, we study the Bayesian framework of the Tensor Network from two perspective. First, we introduce the prior distribution to the weights in the Tensor Network and predict the labels of the new observations by the posterior predictive (marginal) distribution. Since the intractability of the parameter integral in the normalization constant computation, we approximate the posterior predictive distribution by Laplace approximation and obtain the out-product approximation of the hessian matrix of the posterior distribution of the Tensor Network model. Second, to estimate the parameters of the stationary mode, we propose a stable initialization trick to accelerate the inference process by which the Tensor Network can converge to the stationary path more efficiently and stably with gradient descent method. We verify our work on the MNIST, Phishing Website and Breast Cancer data set. We study the Bayesian properties of the Bayesian Tensor Network by visualizing the parameters of the model and the decision boundaries in the two dimensional synthetic data set. For a application purpose, our work can reduce the overfitting and improve the performance of normal Tensor Network model.
## Authors
• 6 publications
• 8 publications
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Driven by the critical needs of biomanufacturing 4.0, we present a proba...
07/08/2018
### BALSON: Bayesian Least Squares Optimization with Nonnegative L1-Norm Constraint
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We introduce a novel framework for the estimation of the posterior distr...
06/25/2018
### Stochastic natural gradient descent draws posterior samples in function space
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## 1 introduction
Bayesian Learning is a framework to combine the internal data information with your background knowledge on your specific learning task, namely the external information in a logically consistent way by Bayes Theorem
draper2006bayesian; draper2013bayesian
. In the Bayesian framework, the parameters of the model are not fixed constants but random variables whose distributions should be introduced according to the background information. From a information perspective, the background information (the knowledge) external to the data set is ’injected’ into the learning model optimally by Bayes Theorem
zellner1988optimal.
In this framework, parameters are not estimated by point estimator such as the Maximum Likelihood Estimator (M.L.E.) instead the posterior distribution of the parameters are given. In this step, the technical difficulty is the computation of the normalization constant. In some application cases, people use the Dirac distribution of the optimal mode to approximate the posterior distribution (M.A.P.) which is equivalent to use the Maximum A Posterior Estimator to do point estimation of the parameters in the model nowlan1992simplifying
. To get better approximation of the posterior distribution, higher order term should be remained and till second order terms are preserved and Normal distribution is obtained by Laplace approximation trick.
In our work, we proposed a robust initialization method to infer the parameters of the Bayesian Tensor Network model more quickly. Since the special structure of the Tensor Network which is a sum of tensor chain products, the output of this model is easily blowing out or decaying to zero which will lead to the instability of the inference process. Using our initialization strategy, the parameters and their gradients will stay in the stable region where gradient descent optimization will behave healthily. To predict the new observations, we get the predictive marginal posterior distribution of Bayesian Tensor Network model by Laplace Approximation trick, namely second order approximation. We observe that the hessian matrix of the Tensor Network model has a nice analytical expression which will be computed more efficiently than the Bayesian Neural Network. Hopefully, the Bayesian Tensor Network can be easily scaled to much larger enterprise level applications.
In practical application, labels will be assigned to the new observations in classification problem. To determine the decision boundary, the utility matrix (negative loss matrix) will be written down and then the decision threshold will be determined by minimize the expected loss.
## 2 Related Work
Deep Neural Networks have made great achievements in recent years lecun2015deep; hinton2006fast; lecun1995convolutional; krizhevsky2012imagenet; hochreiter1997long; srivastava2014dropout. Actually several work were proposed to develop the Bayesian Framework of Neural Network. In the Bayesian Framework of Neural Network, the key part and also difficulty part is to approximate the posterior distribution and also the predictive marginal distribution.
In the framework of hierarchical modeling for model uncertainty which is published by another author of this paper, Prof. David Draper proposed a general framework to consider the model uncertainty and also discussed the computation techniques in inference and prediction steps in the Bayesian Modeling considering model uncertainty draper1995assessment
. At the same time, some other early pioneers of Bayesian Neural Networks also contributes great to this area. David MacKay focus on the Bayesian Neural Networks as an adaptive learning model which means he used the Bayesian Framework to do Neural Networks model comparing and the hyperparameters and model structures selection by the evidence framework
mackay1992interpolation. Radford Neal focus on constructing appropriate priors for the Neural Networks based on the properties of the Neural Networks. Neal suggested to use the infinite hidden units network and the priors converge to the stochastic process. Neal also proposed to use the Hybrid Monte Carlo to do the posterior predictive distribution (integral) simulation neal2012bayesian. Instead of approximate the posterior integral, variational method idea, namely searching in the parameterized functional space by maximizing the Evidence Lower Bound is widely explored blei2006variational; kingma2013auto. Recently, there are amounts of developments following and developing above ideas buntine1991bayesian; tran2019bayesian; lee2017deep; arora2019exact; arora2019harnessing; blundell2015weight; xiong2011bayesian; salakhutdinov2008bayesian; balan2015bayesian; hernandez2015probabilistic.
Tensor Networks are original proposed to describe the quantum many-body states orus2019tensor; chabuda2020tensor; glasser2020probabilistic. In the last several years, several work were proposed such as using the Matrix Product States (MPS) to construct new Learning Models novikov2015tensorizing; stoudenmire2016supervised; han2018unsupervised.
## 3 Initialization of Tensor Network
### 3.1 Background and Set Up
Although Tensor Network has powerful representation ability because of the bond dimension between every pair of tensor nodes neighboring each other, it is difficult to train long Matrix Product States chains. Since every tensor node in the chain will contribute one multiplication factor, the output will roughly increase or decrease with a exponential rate with respect to the number of nodes.
The Set up of the Tensor Network is as follows,
ψl(x)=∑{αi,si}As1α1α2⋯Asilαiαi+1⋯Asnαnα1Φs1⋯sn(x), (1)
where
Φs1⋯sn(x)=ϕs1(x1)⋯ϕsi(xi)⋯ϕn(xn).
We can get a rough estimation of the amplitude of as
|ψl(x)|∼snαnO(Cn), (2)
where
s=|{si}|,α=|{αi}|.
With the assumption that , and , where represents the bond dimension, represents the dimension of the kernel space and represents the number of the nodes.
The networks will not work well or even not work if the tensor nodes are not well initialized since little perturbation of the tensor weights will be amplified because of the exponential rate respect to the number of nodes.
To solve the respond or gradients exploding or vanishing problem, we focused on initialize the parameters with the appropriate distribution. To train deep neural networks, Glorot and Bengio glorot2010understanding
proposed to initialize the parameters with the scaled uniform distribution in the Sigmoid neuron case. For the Relu neuron, He and his collaborators
he2015delving
proposed to use the Gaussian distribution with neural number determined variance. In the following section, we will derive the variance of the Gaussian distribution used in our initialization method.
### 3.2 Variance Analysis
We use the Gaussian distribution to initialize Bayesian Tensor Networks model and set the mean to be zero since we want the respond to stay around zero. For the variance, we need to analyze the variance of the respond as the function of the variance of the tensor weights.
Var[ψl(x)] =Var[∑{s,α}As1α1α2⋯Asilαiαi+1⋯Asnαnα1Φs1⋯sn(x)] (3) =∑{s}αnVar[As]nVar[ϕs(x)]n (4)
In above derivation, we assume that all the components of the tensor weights are independent and identically distributed (I.I.D.) random variables. For the kernel part, we assume that all the components of input kernel are also I.I.D..
Considering the particular kernel function , we have
Var[ψl(x)]=snαnVar[A]nVar[x]n. (5)
From the gradients perspective, we can write down the gradients of tensor node as follows.
[∇ψl(x)]lαiαi+1=∑{s,α}∖{si,αi,αi+1}As1α1α2⋯Asi−1αi−1αiAsi+1αi+1αi+2⋯Asnαn−1αnΦs1⋯si−1si+1⋯sn(x). (6)
We get the variance of the gradients as
Var[∇ψl(x)]=sn−1αn−2Var[A]n−1Var[x]n. (7)
In a good initialization method, we can set the variance of the tensor nodes as the same order of the output variance which will avoid increasing or decreasing the response exponentially, so we get
Var[ψl(x)]∼Var[A]→Var[A]∼O((sα)−nn−1Var[x]−nn−1) (8)
From the gradient perspective, the variance of the gradients is a same order number as the variance of the tensor weights , then we obtain
Var[∇ψl(x)]∼Var[A]O(C)→Var[A]∼O((C1n−2s−n−1n−2α−1Var[x]−nn−2) (9)
We analyze the asymptotic behavior of our initialization formula. From the tensor weights perspective,
Var[A]n→∞=O((sαVar[x])−1). (10)
Var[A]n→∞=O((sαVar[x])−1. (11)
In the large limit, we get the same initialization variance of the tensor weights from weights perspective and gradient perspective which means our initialization method is logically consistent.
Different from Xavier glorot2010understanding, and He initialization he2015delving, our initialization formula does not depend on the number of the nodes heavily as we see the number of nodes just get into the index of the variance of each feature of the data as a fractional factor and the factor will converge to quickly. However, the bond dimension and the physical dimension, namely the dimension of the kernel has a huge effects on the initialization.
We analyse the mean of the parameters following the above idea,
E[ψl(x)] =E[∑{αi,si}As1α1α2⋯Asilαiαi+1⋯Asnαnα1Φs1⋯sn(x)] (12) =snαnE[Asiαi−1αi]nE[ϕ(xi)]n, (13)
In most practical case, the training data set is usually pre-processed and the mean of the training data set is usually transferred to . So we have
E[ϕ(xi)]=0, (14)
then we get
E[ψl(x)]=0. (15)
Similarly, we can get
E[∇ψl(x)]=0. (16)
This means that the mean of the responses and the gradients will always stay in healthy region, namely no matter what the initialization distribution is. So we do not need to care about the mean in the initialization method design.
### 3.3 Numerical Results
In this section. we study the performance of our formula on the MNIST lecun1998mnist, Phishing Website and Breast Cancer data set Dua:2019 and also we compare our initialization method with Xavier and He initialization methods.
In Fig. 1, we compare the accuracy on the MNIST Data set in the first epochs by our initialization method, Xavier initialization method and He initialization method. Our initialization method converges much more quickly than the other two initialization method on the MNIST data set.
In Fig. 0(a), Fig. 0(c) and Fig. 0(e), we compare the convergence of the Bayesian Tensor Network with our initialization method, Xavier initialization and He initialization method on the MNIST data set, Phishing data set and Breast Cancer data set. In these three data set, our initialization method works much better than the other two methods. Since the number of features in the Breast Cancer data set is only , so the training of the Bayesian Tensor Network is not as the same heavily sensitive to the initialization method as the Phishing data set or the MNIST data set.
In Fig. 0(b), Fig. 0(d), and Fig. 0(f), we show the accuracy curves with the standard deviation obtained by our formula, and slightly scaled std deviation. In the data set whose data has more number of features, the training process is more sensitive to the initialization and small deviation from the std given by our formula will lead to the bad convergence.
## 4 Bayesian Framework For Tensor Networks
### 4.1 The General Framework
We write down our Bayesian Tensor Networks model as follows,
(17)
In the inference step, we write down the posterior distribution of the parameters , namely as
p(A|D,α,B)∝p(D|A)p(A|α,B). (18)
If we use the Maximum A Posterior Estimator (M.A.P.) to estimate the parameters in the optimal mode of the model which is the same as the normal Tensor Network model,
AMAP=argmaxAlogp(A|D,α,B). (19)
In our work, we focus on the Full Bayesian Analysis. We do prediction and make decision based on the predictive posterior marginal distribution.
p(^t|^x,D,α,B)=∫p(^t|^x,A)p(A|D,α,B)dA (20)
However, since the analytical intractability of the predictive posterior marginal distribution, we approximate the marginal predictive distribution around the M.A.P. mode of the posterior distribution till second order. We note that if we approximate the posterior distribution roughly by the Dirac distribution at the M.A.P. mode,
we get
p(^t|^x,D,α,B)≃p(^t|^x,AMAP,α,B). (21)
Obviously the information contained in the data set is lost in the point approximation.
According to the predictive posterior marginal distribution obtained in the prediction step, we can assign every new observation in the test set with one label which is a decision problem. For the goal of minimizing the misclassification rate in the classification problem, the decision boundary is determined as
C=argmax^tp(^t|D,α,B) (22)
For some practical case where utility matrix (negative loss matrix) is specially designed, we can also maximize the expected utility function to determine the action. For the regression problem, we focus on the mean square loss and by minimize the expected loss, it is proved that the decision boundary is the conditional expectation of the label given the features, namely , which is just the value of the regression function .
### 4.2 Classification
In our convention, the data set is notated as
D={(x(i),t(i))|i∈(1,⋯,m),x(i)∈R(n),t(i)∈{1,2,⋯,k}}, (23)
in the classification problem. In our set up,
is encoded as the one-hot vector which represents which category the data
belongs to.
For the binary classification problem, the response of the Bayesian Tensor Network model is the logits, namely
logp(t(i)|x(i),A)1−p(t(i)|x(i),A)=ψ(x(i),A). (24)
If we treat every component of the encoded vector independently and model each component with above logit formula, then it can be easily extended to multi-classification case.
In the multi-classification case, we use the Softmax activation function and we get
p(t(i)j|x(i),A)=% softmax(ψj(x(i),A)). (25)
For the binary case, we can write down the cost function as
L(A)∼−m∑i(t(i)logσ(ψ(A,x(i)))+(1−t(i))log(1−σ(ψ(A,x(i)))))+12α|A|2. (26)
For the multi-classification case, we have
L(A)∼−∑ij(ψ(i)j+log(∑jψ(i)j))+12α|A|2. (27)
In the Bayesian Tensor Network, we do not need to do inference (training) if we can solve the intractable posterior marginal distribution as long as the the prior distribution is wisely introduced according to the background knowledge. However, we need to find the M.A.P mode of the posterior distribution to expand the posterior distribution around the M.A.P mode. We write down the posterior distribution and use the stochastic gradient optimization method to get to the M.A.P. mode. In practice, our objective function is the negative log posterior distribution. Around the optimal mode, we get the normal distribution as
p(A|D,α,B)≃q(A|D,α,B), (28)
where
The Hessian matrix contains the geometric information (curvature) of the posterior distribution, so more information is extracted by the Hessian matrix which is a better approximation than the distribution.
The co-variance matrix of the Normal distribution is
M =∂2∂A2logp(A|D,α,B)=H+1αI, (29)
where is the second derivative matrix of the log likelihood function . The time complexity of computing the inverse of the Hessian matrix is which is time consuming, so we use the Out-Product approximation to decrease the time complexity to .
Since the prior distribution and the approximated posterior distribution are all Normal distribution, then we can get the predictive marginal posterior distribution analytically as
p(^t|^x,D,α) =∫p(^t|^x,A)p(A|D,B)dA (30) ≃∫p(^t|^x,A)q(A|D,B)dA (31)
By plugging in the approximated posterior distribution , we get
p(^tj|^x,D,α)=σ(κ(σ2j)μ′j), (32)
where
σ2j=∇ψjMAPM−1∇ψjMAP, μ′j=ψjMAP−log∑−jexp(ψjMAP).
### 4.3 Hessian Matrix
The approximation of the Hessian matrix has been widely studied. In the Out-Product approximation method, the idea is in the trained networks, the label and the output are close to each other then the second derivative matrix term is very small which is ignored. We get the out-product approximation of the Hessian matrix of the Bayesian Tensor Network model as
H=∑i,jt(i)j(∑k,k′y(i)ky(i)k′∂ψ(i)k∂A∂ψ(i)k′∂A−∑ky(i)k(∂ψ(i)k∂A)2) (33)
Our result above just contains the first derivative which means the time complexity is almost . Here means the component of the output of the Bayesian Tensor Networks. Different from the Neural Networks, the first derivative of the logits can be calculated analytically and then we can get an analytical result of the Hessian matrix of the Bayesian Tensor Networks.
### 4.4 Numerical Results
We study the performance of the Bayesian Tensor Networks on several data set.
From small data set to big data set, we used the following data sets
• Synthetic Data Set: Two dimensional Gaussian Blobs with two classes.
• Breast Cancer Wisconsin Data Set: A toy binary classification data set.
• Phishing Website Data Set: A small binary classification data set.
• MNIST Data Set: A standard multi-classification data set in computer vision community.
To study the Bayesian effects, we visualize the parameters in the Bayesian Tensor Network and the decision boundary in two dimensional synthetic data set.
We study the performance of the Bayesian Tensor Network with different standard deviation and different bond dimension on the Breast Cancer Wisconsin, Phishing Website and MNIST data set.
We train the Bayesian Tensor Networks and Bayesian Neural Networks on the blobs synthetic data set which contains samples in two classes. We used relatively bigger tensor nets and neural nets model to overfit the data set to study the Bayes shrinkage effect with different prior distribution in Bayesian Tensor Network and Bayesian Neural Network in Fig. 2. As we use greater standard deviation in the prior Normal distribution, the decision boundary becomes smoother. From our numerical experiments, we find that neural network is slightly more sensitive to the prior distribution than the tensor network.
In Fig. 3, we show the histogram of the parameters in the trained Bayesian Tensor Network and Bayesian Neural Network. We observe the Bayesian shrinkage in both the histograms of the parameters in the Bayesian Tensor Network and the Bayesian Neural Network. In the Bayesian Tensor Network, we find that the distribution of the parameters is not heavily affected instead the stand deviation is decreased as the prior std deviation is decreased. For the Bayesian Neural Network, we find that the Bayesian shrinkage effect is heavier and the parameters distribution gets to be heavy tail which means the parameters in the model become sparse.
Bond dimension is a key hyperparameter in the MPS which controls the ’description’ ability of the model collura2019descriptive. In Fig. 4, we show the test accuracy of the Bayesian Tensor Network model with different bond dimension in different data set. We observe that as the bond dimension gets increased, the generalization ability of the model becomes better, namely the Bayesian Tensor Network model gets better prediction accuracy.
## 5 Conclusion
We study the Bayesian framework of the Tensor Network and propose a robust initialization method. We use the toy, small and standard data set: Breast Cancer, Phishing website and MNIST data set to evaluate our initialization method and study the performance of the Bayesian Tensor Network model. We observe the Bayesian shrinkage in the parameters histogram plot and study the decision boundary of the Bayesian Tensor Network. We also explore the bond dimension in the Bayesian Tensor Network model. In practical application, we expect our model to take its own advantage in the small data set where overfitting problem can be solved by prior information introducing.
## Acknowledgements
The authors wish to thank David Helmbold, Hongyun Wang, Qi Gong, Torsten Ehrhardt and Francois Monard for their helpful discussions.
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2022-05-18 03:04:39
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https://mathematica.stackexchange.com/questions/189289/edo-with-parameter-divergence-and-stop-integration
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# EDO with parameter, divergence and stop integration
I'm trying to solve an ODE which independent variable is time, where there is a parameter involved, let's say $$\beta$$. I need to plot the final solution in terms of $$\beta$$. The equations, initial conditions and the code I used are $$\begin{eqnarray} &&\dot{y}=\sin(\beta)\,x^2\\ &&\dot{x}=y\\ &&x(0)=5\\ &&y(0)=\tan(\beta)*x(0) \end{eqnarray}$$
sol = ParametricNDSolve[{D[y[t], t] == Sin[\[Beta]]*x[t]^2,D[x[t], t] == y[t], x[0] == 5, y[0] == Tan[\[Beta]]*x[0]}, {y,x}, {t, 0, 20}, {\[Beta]}];
Plot[Evaluate[y[\[Beta]][0.9] /. sol], {\[Beta], 0, 2}]
This is the plot of y in terms of $$\beta$$. Apparently, the solution diverges for some $$\beta$$ and for each time the graphics changes. How can I stop the integration when x or y grows too much, let's say 2000, save the time that it happens and then plot the final solution in terms of $$\beta$$ in this certain time?
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2019-01-22 09:27:32
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5419066548347473, "perplexity": 474.40581537398964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583831770.96/warc/CC-MAIN-20190122074945-20190122100945-00139.warc.gz"}
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http://physics.stackexchange.com/questions/28314/how-should-i-be-thinking-about-tides
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# How should I be thinking about tides?
I am working on a project for physics that involves tides. This is my current mind set when thinking about tides:
The earth applies a gravitational force on some mass M. The moon & sun apply a gravitational forces on the mass M away from the earth. The bodies of water on the earth have some acceleration towards the moon & sun due to their forces (F=ma). The bodies of water accelerate till they reach a sort of equilibrium between forces.
I would like you to poke holes in my understanding of tides and try and answer my naive questions bellow.
How would you describe tides effected by the moon using forces? How can that be used to calculate the height of tides?
-
Don't ask Bill O'Reilly. He was completely stumped by tides, and attributed them to an act of God. Lol! – ja72 May 15 '12 at 5:34
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2015-05-05 04:15:22
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http://www.openmath.org/cd/s_data1.xhtml
|
Home Overview Documents Content Dictionaries Software & Tools The OpenMath Society OpenMath Projects OpenMath Discussion Lists OpenMath Meetings Links
# OpenMath Content Dictionary: s_data1
Canonical URL:
http://www.openmath.org/cd/s_data1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
s_data1.ocd
CD as XML Encoded OpenMath:
s_data1.omcd
Defines:
mean, median, mode, moment, sdev, variance
Date:
2004-03-30
Version:
3
Review Date:
2006-03-30
Status:
official
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work. Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society. In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such. This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
compilation of derived works, but keep paragraphs a) and b)
intact. The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
society at http://www.openmath.org.
This CD holds the definitions of the basic statistical functions used on sample data. It is intended to be `compatible' with the MathML elements representing statistical functions, though it does not cover the concept of random variable which is mentioned in MathML.
## mean
Role:
application
Description:
This symbol represents an n-ary function denoting the mean of its arguments. That is, their sum divided by their number.
Commented Mathematical property (CMP):
The mean of n arguments is their sum divided by their number
Formal Mathematical property (FMP):
$\mathrm{apply_to_list}\left(\mathrm{mean},L\right)=\frac{\mathrm{apply_to_list}\left(+,L\right)}{\mathrm{size}\left(L\right)}$
Example:
The mean of {1,2,3} is 2
$\mathrm{mean}\left(1,2,3\right)=2$
Signatures:
sts
[Next: sdev] [Last: moment] [Top]
## sdev
Role:
application
Description:
This symbol represents a function requiring two or more arguments, denoting the sample standard deviation of its arguments. That is, the square root of (the sum of the squares of the deviations from the mean of the arguments, divided by the number of arguments). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, (7.7.11) section 7.7.1.
Commented Mathematical property (CMP):
The square of the standard deviation of n arguments is the sum of the squares of the differences from their mean divided by the number of arguments.
Formal Mathematical property (FMP):
${\mathrm{apply_to_list}\left(\mathrm{sdev}\left(L\right)\right)}^{2}=\frac{\mathrm{apply_to_list}\left(+,\mathrm{list}\left({\left(x-\mathrm{mean}\left(L\right)\right)}^{2}|x\in L\right)\right)}{\mathrm{size}\left(\right)}$
Example:
This is an example to denote the standard deviation of a set of data
$\mathrm{sdev}\left(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts
[Next: variance] [Previous: mean] [Top]
## variance
Role:
application
Description:
This symbol represents a function requiring two or more arguments, denoting the variance of its arguments. That is, the square of the standard deviation.
Commented Mathematical property (CMP):
The variance of n arguments is the square of the standard deviation of those arguments.
Formal Mathematical property (FMP):
$\mathrm{apply_to_list}\left(\mathrm{variance}\left(L\right)\right)={\mathrm{apply_to_list}\left(\mathrm{sdev}\left(L\right)\right)}^{2}$
Example:
This is an example to denote the variance of a set of data
$\mathrm{variance}\left(3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts
[Next: mode] [Previous: sdev] [Top]
## mode
Role:
application
Description:
This symbol represents an n-ary function denoting the mode of its arguments. That is the value which occurs with the greatest frequency.
Commented Mathematical property (CMP):
The mode of n arguments is that value which occurs with the greatest frequency.
Example:
The mode of {1,1,2} is 1
$\mathrm{mode}\left(1,1,2\right)=1$
Signatures:
sts
[Next: median] [Previous: variance] [Top]
## median
Role:
application
Description:
This symbol represents an n-ary function denoting the median of its arguments. That is, if the data were placed in ascending order then it denotes the middle one (in the case of an odd amount of data) or the average of the middle two (in the case of an even amount of data).
Example:
The median of {1,2,3} is 2
$\mathrm{median}\left(1,2,3\right)=2$
Signatures:
sts
[Next: moment] [Previous: mode] [Top]
## moment
Role:
application
Description:
This symbol is used to denote the i'th moment of a set of data. The first argument should be the degree of the moment (that is, for the i'th moment the first argument should be i), the second argument should be the point about which the moment is being taken and the rest of the arguments are treated as the data. For n data values x_1, x_2, ..., x_n the i'th moment about c is (1/n) ((x_1-c)^i + (x_2-c)^i + ... + (x_n-c)^i). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1.
Example:
This is an example to denote the 2'nd moment of a set of data about the origin.
$\mathrm{moment}\left(2,0,3.1,2.2,1.8,1.1,3.3,2.4,5.5,2.3,1.7,1.8,3.4,4.0,3.3\right)$
Signatures:
sts
[First: mean] [Previous: median] [Top]
Home Overview Documents Content Dictionaries Software & Tools The OpenMath Society OpenMath Projects OpenMath Discussion Lists OpenMath Meetings Links
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2015-05-27 11:30:50
|
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https://math.stackexchange.com/questions/4180043/differentiation-of-a-piecewise-function-at-a-point/4180059
|
# differentiation of a piecewise function at a point
say I have the function $$f(x)=\begin{cases} x^{3}\sin(\frac{5}{x}) & x\ne0\\ 0 & x=0 \end{cases}$$ I want to prove it is differentiable at $$0.$$ I first show that it is continuous by: $$\lim_{x\to0}f(x)=x^{3}\sin(\frac{5}{x})=0$$ That is because I have a bounded function multiplied by 0. Now i need to show if f is differentiable and if so,is $$f'$$ continuous? I started by doing $$f'(0)=\lim_{h\to0}\frac{(0+h)^{3}\sin(\frac{5}{0+h})-(0)^{3}\sin(\frac{5}{0})}{h}$$ But I can't seem to understand how do I go about the $$\sin (\frac{5}{0})$$ part because that doesn't exist. Do i take 2 limits of $$x\to 0$$ and $$h\to 0$$ in that case? As for the second part of showing that f' is continuous I just differentiated with the product rule and got that: $$f'(x)=3x^2\sin(\frac{5}{x})-5x\cos(\frac{5}{x})$$ And after checking both sides I saw that one of them is negative and the other is positive, hence it has different limits at the two sides. How do I solve these problems by definition?
• Why did you write $0^3\sin\frac 50$? The hypothesis does not say that $f(0)=0^3\sin\frac 50$.
– user239203
Jun 22 '21 at 15:30
• So instead I just say that $$f'(0)=\lim_{h\to0}\frac{(0+h)^{3}\sin(\frac{5}{0+h})-0}{h}=\frac{h^{3}sin(\frac{5}{h})}{h}=h^{2}\sin(\frac{5}{h})=0$$ Jun 22 '21 at 15:39
• @DannyBlozrov Yes, that is the correct way to evaluate $f'(0)$. Jun 22 '21 at 15:49
The defintion of the derivative at $$0$$ is :$$f'(0)=\lim_{h\to0}\frac{f(0+h)-f(0)}{h}$$ You are given that $$f(0)=0$$ from here the limit is easy to evaluate.
For other $$x$$ you can use the product and chain rule .
• (+1) for a complete solution that actually addresses the OP's first question. Jun 22 '21 at 15:48
You have that $$\begin{eqnarray*} f: I\subset \mathbb{R} &\to& \mathbb{R}\\ x&\mapsto& f(x)=\begin{cases}x^{3}\sin\left(\frac{5}{x}\right), &\quad x\not=0\\ 0, &\quad x=0 \end{cases} \end{eqnarray*}$$ If you want to prove that $$f$$ is differentiable at $$0$$, you do not need to start by proving that $$f$$ is continuous at $$0$$. Of course, if $$f$$ is not continuous at $$0$$, then $$f$$ is not differentiable at $$0$$. But, it is not what is requested in the problem.
You need to prove that $$\lim_{h\to 0} \frac{f(0+h)-f(0)}{h}$$there exists and if that limit there exists, then $$f'(0)=\lim_{h\to 0} \frac{f(0+h)-f(0)}{h}.$$
Now, you know that if $$x=0$$, so $$f(0)=0$$ and for all $$x\not=0$$, you have that $$f(x)=x^{3}\sin\left(\frac{5}{x}\right)$$.
Hence, $$\lim_{h\to 0}\frac{h^{3}\sin\left(\frac{5}{h}\right)-0}{h}=\lim_{h\to 0}h^{2}\sin\left(\frac{5}{h}\right)=0. \quad (1)$$
Note that $$(1)$$ it's clear, because $$-1\leqslant \sin\left(\frac{5}{h}\right)\leqslant 1$$ $$-h^{2}\leqslant h^{2}\sin\left(\frac{5}{h}\right)\leqslant h^{2}$$ $$\lim_{h\to 0} -h^{2}\leqslant \lim_{h\to 0} h^{2}\sin\left(\frac{5}{h}\right)\leqslant \lim_{h\to 0} h^{2}$$ $$0 \leqslant h^{2}\sin\left(\frac{5}{h}\right)\leqslant 0$$ Therefore, $$\lim_{h\to 0} h^{2}\sin\left(\frac{5}{h}\right)=0$$ and $$f$$ is differentiable at $$x=0$$ with $$f'(0)=0$$.
• (+1) for a complete solution that actually addresses the OP's first question. Jun 22 '21 at 15:48
• Thank you very much,understood it all. As for showing that f' is continuous I can just use the product and chain rule to find $f'(x) for any$ x\ne 0 $and I know that$ f'(0)=0 \$ ,now I just have to compare the limit to 0 and see if its the same? Jun 22 '21 at 16:01
To check differentiability at $$0$$ you should essentially calculate the limit $$\lim_{h \to 0} \frac{f(h) -f(0)}{h} = \lim_{h\to 0} \frac{h^3 \sin(5/h)}{h} = \lim_{h \to 0} h^2 \sin \left( \frac{5}{h} \right)$$.
Observe that $$0 \le \left\vert h^2 \sin \left( \frac{5}{h} \right) \right\vert \le h^2$$ So by Squeeze_theorem the limit exists and equal to $$0$$.
• Not really following why, that's a must have condition that checks if it is continuous, why does it mean its differentiable? Jun 22 '21 at 15:44
• This fails to address the main question. Jun 22 '21 at 15:47
When you look at the product rule, you need that both factors are differentiable. But $$\sin(5/x)$$ is not differentiable in $$0$$, so you cannot apply this rule to your example.
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2022-01-22 07:25:56
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https://math.stackexchange.com/questions/929882/what-is-wrong-with-my-algorithm-for-finding-how-many-positive-integers-are-divis
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# What is wrong with my algorithm for finding how many positive integers are divisible by a number d in range [x,y]?
I have been solving basic counting problems from Kenneth Rosen's Discrete Mathematics textbook (6th edition). These come from section 5-1 (the basics of counting), pages 344 - 347.
This question is not specifically about finding an answer to a problem or being given the correct equation, but whether my reasoning is sound. Therefore I would find it hard to argue this is a duplicate of seemingly similar questions like this one or this one.
The problems I have been dealing with come of the form How many positive integers in range [x,y] are divisible by d? All additional questions are based on the composition of the information learned in these, e.g. how many positive integers in range [x,y] are divisible by d or e?
To answer the simple question I wrote this "equation/algorithm," which takes as input an inclusive range of positive integers $[x,y]$ and a positive integer $d$, and returns $n$, the total number of positive integers in range $[x,y]$ which are divisible by $d$.
(1) $n = \left \lfloor{\frac{y}{d}}\right \rfloor - \left \lfloor{\frac{x}{d}}\right \rfloor$
The idea is that in order to count how many positive integers are divisible by $d$ from $[1,m]$, we simply calculate $\left \lfloor{\frac{m}{d}}\right \rfloor$, because every $dth$ positive integer must be divisible by $d$. However, this does not work when given a range $[x,y]$ where $x \not= 1$ or when $x > 1$. So we need to subtract the extra integers we counted, which is $\left \lfloor{\frac{x}{d}}\right \rfloor$, i.e. the number of positive integers divisible by $d$ from $[1,x]$.
For a sanity check, I also wrote a brute force algorithm that does a linear search over every positive integer in the range $[x,y]$ and counts it if $x \text{ mod } d == 0$. It also can list out the integers it picked, in case I am feeling really paranoid.
With (1) I've been getting the correct answers except on this problem/input: How many positive integers between 100 and 999 inclusive are odd? My solution was to calculate how many are even, and subtract this from the total number of positive integers in range $[100,999]$. To find the evens I simply use the algorithm in (1):
$\left \lfloor{\frac{999}{2}}\right \rfloor - \left \lfloor{\frac{100}{2}}\right \rfloor = 499 - 50 = 449$
But this answer is wrong, since there actually $450$ even numbers in range $[100,999]$ by the brute force algorithm. (1) is somehow counting off by 1. My question is, why is (1) failing for this input of $(2, [100,999])$ but so far it's worked on every other input? What do I need to do to fix (1) so it produces the correct answer for this case? Perhaps I'm actually over counting because $x$ should actually be $x - 1$?
(1') $n = \left \lfloor{\frac{y}{d}}\right \rfloor - \left \lfloor{\frac{x - 1}{d}}\right \rfloor$
(1') returns the correct answer for this specific input now, but I am not sure if it will break my other solutions.
After computing the number of positive multiples of $d$ less than or equal to $y,$ you correctly want to subtract the multiples that are not actually in the range $[x,y].$ Those are the multiples that are less than $x.$ When $x$ is divisible by $d,$ then $\left\lfloor \frac xd \right\rfloor$ counts all multiples of $d$ up to and including $x$ itself. So as you surmised, you want to subtract $\left\lfloor \frac {x-1}d \right\rfloor$ instead. This is true for any divisor, not just $d=2.$
Suppose you have $$(k-1)d<x\leq kd<(k+1)d<\ldots<(k+n)d\leq y<(k+n+1)d.$$ Then, you always have $\lfloor{\frac{y}{d}}\rfloor=k+n$ is $k+n$. On the other hand, if $x$ is divisible by $d$, then $\lfloor{\frac{x}{d}}\rfloor=k$, whereas if $x$ isn't divisible by $d$, then $\lfloor{\frac{x}{d}}\rfloor=k-1$. To rectify this, instead use $\lceil{\frac{x}{d}}\rceil$ which always returns $k$. Your solution is then $$n+1=(n+k)-k+1=\color{blue}{\left\lfloor{\frac{y}{d}}\right\rfloor-\left\lceil{\frac{x}{d}}\right\rceil+1}.$$
p.s. $$(k-1)d<x\leq kd\implies(k-1)d\leq x-1<kd\implies\left\lfloor{\frac{x-1}{d}}\right\rfloor=k-1$$ so that $$n+1=(n+k)-(k-1)=\color{red}{\left\lfloor{\frac{y}{d}}\right\rfloor-\left\lfloor{\frac{x-1}{d}}\right\rfloor}.$$ The formulas in blue and red produce the same answer.
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2020-01-27 03:07:31
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https://artofproblemsolving.com/wiki/index.php?title=1991_AHSME_Problems/Problem_1&diff=prev&oldid=56404
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# Difference between revisions of "1991 AHSME Problems/Problem 1"
If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is
(A) $-2$ (B) $-\frac{2}{5}$ (C) $-\frac{1}{4}$ (D) $\frac{2}{5}$ (E) $2$ The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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2021-09-17 03:35:19
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https://blog.bordes.me/google-ctf-2018-mitm.html
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# Google CTF 2018: Crypto - MITM
Posted on 01 Jul 2018, 10:00 in WriteUps
## Challenge description
The challenge was called MITM and as confirmed by the small description given ("Man in the Middle the communication between the client and the server.") we will need to execute a Man-In-The-Middle (MITM) attack There are two more informations given: a zip file containing a python source code and a netcat command line (nc mitm.ctfcompetition.com 1337) to reach the challenge-server where the code is running.
Before trying to find where is the flaw, the general approach used is to answer the following questions:
• What kind of python modules are used?
• What is the role of each function?
• How does the interaction with the challenge-server work?
### Python modules used
The source code starts by importing a bunch of modules. Some are just helpers to manage input and outputs (not really interesting):
from binascii import hexlify
from binascii import unhexlify
import logging
import sys
import os
And some are modules implementing cryptographic primitives:
from curve25519 import Private, Public
import nacl.secret
import hmac
import hashlib
After a little bit of research in the source code, hmac and hashlib will be used to define a HMAC based on SHA256, nacl.secret to ensure confidentiality of the data sent between the server and the client (based on a share secret), and curve25519to established this shared secret.
### Roles of functions
Apart from helper functions (ReadLine, WriteLine, ReadBin, WriteBin) that are used to manage input and outputs, their are 6 main functions.
#### ComputeProof
Python source code
def ComputeProof(key, data):
return hmac.new(key, data, digestmod=hashlib.sha256).digest()
Takes a key and some data as input. Returns the HMAC of the data under the key.
#### VerifyProof
Python source code
def VerifyProof(key, data, proof):
return hmac.compare_digest(ComputeProof(key, data), proof)
Takes a key, some data and a proof as input. Returns the result of the verification proof (checks that the proof and the HMAC of the data under the key are equals).
#### Handshake
Python source code
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 def Handshake(password, reader, writer): myPrivateKey = Private() myNonce = os.urandom(32) WriteBin(writer, myPrivateKey.get_public().serialize()) WriteBin(writer, myNonce) theirPublicKey = ReadBin(reader) theirNonce = ReadBin(reader) if myNonce == theirNonce: return None if theirPublicKey in (b'\x00'*32, b'\x01' + (b'\x00' * 31)): return None theirPublicKey = Public(theirPublicKey) sharedKey = myPrivateKey.get_shared_key(theirPublicKey) myProof = ComputeProof(sharedKey, theirNonce + password) WriteBin(writer, myProof) theirProof = ReadBin(reader) if not VerifyProof(sharedKey, myNonce + password, theirProof): return None return sharedKey
This function takes a pre-shared password and establish a new shared secret between the caller of the function and another entity (server or client). This handshake is made through a supposed insecure channel and is based on Curve25519. Curve25519's goal is to provide a Diffie-Hellman protocol based on an elliptic curve: each entity generate a random private key PrivA (respectively, PrivB) and derives a public key PubA (respectively, PubB) that are used to generate a secret S = f(PubA, PrivB) = f(PubB, PrivA). This way, only PubA and PubB are send through the insecure channel.
The handshake function used can be described by this diagram:
As seen on this diagram, the Handshake function is completely symmetric. Particularly, the challenge-server executing the Handshake function (either as a client or a server) checks the public key and the nonce sent to him (see the source code, line 11 and 13). The Handshake function fails if:
• The nonce receive is the same as the nonce sent, or;
• The public key received is all null bytes or is 1 represent in little endian.
As we will see in the solution of the challenge, the check of the public key received is a big hint.
#### Server
Python source code
def Server(password, flag, reader, writer):
if sharedKey is None:
WriteLine(writer, b'Error: nope.')
return 1
mySecretBox = nacl.secret.SecretBox(sharedKey)
WriteBin(writer, mySecretBox.encrypt(b"AUTHENTICATED"))
while 1:
if cmd == b'help':
rsp = b'help|exit|whoami|getflag'
elif cmd == b'exit':
return 0
elif cmd == b'whoami':
rsp = b'root'
elif cmd == b'getflag':
rsp = flag
else:
return 1
WriteBin(writer, mySecretBox.encrypt(rsp))
After negotating a new shared secret with a client using Handshake (with the pre-shared password), every communication to this client is encrypted using this shared secret. If the handshake fails (i.e., returns None), it sends an error message in clear.
The server sends 'AUTHENTICATED' to the client to notify him that the handshake ended successfully. It then accepts some basic commands from the client, including getflag that tells him to send the flag.
#### Client
Python source code
def Client(password, reader, writer):
if sharedKey is None:
WriteLine(writer, b'Error: nope.')
return 1
mySecretBox = nacl.secret.SecretBox(sharedKey)
if line != b"AUTHENTICATED":
WriteLine(writer, b'Error: nope.')
return 1
WriteBin(writer, mySecretBox.encrypt(b"whoami"))
if line != b'root':
return 1
WriteBin(writer, mySecretBox.encrypt(b"exit"))
return 0
After negotating a new shared secret with a server using Handshake(with the pre-shared password), every communication to this server is encrypted using this shared secret. If the handshake fails (i.e., returns None), it sends an error message in clear.
The client expect to receive 'AUTHENTICATED' by the server, which is the sign that the handshake ended successfully. Then it sends some commands to the server and checks the output.
#### Challenge
Python source code
def Challenge(password, flag, reader, writer):
try:
is_server = server_or_client[0] in b'sS'
is_client = server_or_client[0] in b'cC'
if is_server:
elif is_client:
else:
WriteLine(writer, b'Error: Select if you want to speak to the (s)erver or (c)lient.')
return 1
except Exception as e:
WriteLine(writer, b'Error')
return 1
This function is used to communicate with the challenge-server. It is this function that is called by the challenge-server when a TCP connection is established. First, the challenge-server expects an input 's' or 'S' (respectively, 'c' or 'C') representing the fact that the player want to talk to the server (respectively, to the client) and be in the role of the client (respectively, the server). According to this input, the challenge-server calls the corresponding function (complementary role to the one chosen by the ctf-player).
## Proposed solution
Now we will see how I manage to capture the flag for this challenge as well as some explanations on how I found this solution.
### Basic and unsuccessful attempts
As often with CTF challenges, the biggest hint is the title and description of the challenge. Here we know that we have to make a man-in-the-middle attack on this protocol. Previously we identified the protocol as being a Diffie-Hellman key exchange. The simple Diffie-Hellman key exchange can be quickly summarize by this simple diagram:
The Diffie-Hellman protocol is vulnerable to a simple man-in-the-middle attack if it is not properly protected by an authentication mechanism. This diagram illustrates this attack:
At the end, Alex (respectively, Billie) think that the communication with Billie (respectively, Alex) is secure but Charlie can decrypt, read and re-encrypt the messages at will.
But in our case the proof mechanism (secured by a state-of-the-art HMAC) is used to authenticate Alex toward Billie and Billie toward Alex using a pre-shared secret (a password) and using a nonce to avoid replays (sending an intercepted previous proof as a proof for the current session). Thus, the classic man-in-the-middle attack on Diffie-Hellman cannot be used.
The handshake being perfectly symmetric, the attacker (as a client or as a server) can wait that the challenge-server sends him data before continuing. Thus an attack like the one describe in the following diagram can theoretically be made.
But because the nonce receive is checked to be different than the one receiveed, this attack will not work in our case.
Even if these attempts were unsuccessful, they are useful to understand why the challenge is not trivial and to come up with an attack that actually works.
### Successful attack
One thing was odd in the source code of the Handshake function: during the handshake, the public key received is checked. I started to wonder why is that, so I juste ran the following code to check what was the shared secret generated by the Curve25519 algorithm when used with the public key rejected by Handshake.
from curve25519 import Public, Private
from binascii import hexlify
weird_pubkey1 = b'\x00'*32
weird_pubkey2 = b'\x01' + b'\x00'*31
def sharedsecret(pubkey):
pubkey_obj = Public(pubkey)
privkey_obj = Private() # Generate a random new private key each time
return privkey_obj.get_shared_key(pubkey_obj)
for i in range(5):
print(hexlify(sharedsecret(weird_pubkey1)))
print(hexlify(sharedsecret(weird_pubkey2)))
The result was surprising:
$python2.7 test.py 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a 68b59f127c671255346e099c3b9ea067a5595ba2adf26daa5e69d6a8a29d191a Even if the private key changes every time, the computed shared secret is the same. That's a good thing that Handshake reject them! But wait, there is more: according to the website of Curve25519 algorithm, the public key checked are not the only 'bad' public key. And these other 'bad' public keys are not rejected by Handshake. We have our vulnerability: we can force the shared secret to be equal to a known value, independently from the value of the private key. Thus, the following attack is possible: We establish with both the server and the client the same known shared secret by using a 'bad' public key with both of them. Then, we use Alex to provide a correct proof using the unknown password and the nonce chosen by the server. Then we send to the server the valid proof (because the shared secret between us and the client and between us and the server are the same). Thus, we can now send commands to the server as if we were the client. By reusing some input-output functions, it is easy to come up with a python program implementing the attack. Here is mine, letting me interact with the server (and send him arbitrary commands) as if I was the client and without knowing the pre-shared secret (password). Python source code of solver.py 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 from challenge import WriteBin, ReadBin from curve25519 import Private, Public import os from binascii import hexlify, unhexlify import nacl.secret import socket weakPublic = Public(b'\xe0\xebz|;A\xb8\xae\x16V\xe3\xfa\xf1\x9f\xc4j\xda\t\x8d\xeb\x9c2\xb1\xfd\x86b\x05\x16_I\xb8\x00') class Reader(): def __init__(self, socket): self.s = socket def read(self, size): return self.s.recv(size) class Writer(): def __init__(self, socket): self.s = socket def write(self, text): return self.s.send(text) def flush(self): None def InitSocket(toWho): s = socket.socket() s.connect(("mitm.ctfcompetition.com",1337)) s.send(toWho+'\n') return (Writer(s), Reader(s)) def Attack(): (writerA, readerA) = InitSocket('c') #(writerA, readerA) = InitSocket('s') (writerB, readerB) = InitSocket('s') myNonce = os.urandom(32) WriteBin(writerB, weakPublic.serialize()) WriteBin(writerB, myNonce) pubkeyB = ReadBin(readerB) nonceB = ReadBin(readerB) proofFromB = ReadBin(readerB) WriteBin(writerA, weakPublic.serialize()) WriteBin(writerA, nonceB) pubkeyA = ReadBin(readerA) nonceA = ReadBin(readerA) proofFromA = ReadBin(readerA) proofToB = proofFromA WriteBin(writerB, proofToB) sharedSecret = Private().get_shared_key(weakPublic) mySecretBox = nacl.secret.SecretBox(sharedSecret) line = mySecretBox.decrypt(ReadBin(readerB)) print(line) while True: cmd = raw_input('Command to send: ') WriteBin(writerB, mySecretBox.encrypt(cmd)) line = mySecretBox.decrypt(ReadBin(readerB)) print(line) Attack() After running the code: $ python2.7 solver.py
AUTHENTICATED
Command to send: whoami
root
Command to send: getflag
CTF{kae3eebav8Ac7Mi0RKgh6eeLisuut9oP}
Command to send: exit
### Remarks
• For Curve25519, the public keys that are considered 'bad' are the one corresponding to points with the x-coordinate equal to 0, 1, -1, 325606250916557431795983626356110631294008115727848805560023387167927233504, or 39382357235489614581723060781553021112529911719440698176882885853963445705823. The x-coordinate is taken modulo 2^555 - 19 and every 256-bits strings corresponds to a point, there are in fact 12 different 'bad' public key strings;
• Even if here it had no impacts, it is generally a bad practice to use the same value as key to compute the HMAC and to encrypt/decrypt further communications;
• The attack only uses the client as a way to compute the needed proof. This proof is computed in the Handshake function, which is exactly the same when used as a server or as a client. Thus, it doesn't matter if we use the server or the client to compute this proof as shown by uncommenting line 38 of the solver program.
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2019-07-21 05:33:50
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http://elib.dlr.de/66665/
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DLR Portal Home | Imprint | Contact | Deutsch
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# Dynamics of large-scale structures and heat transfer in turbulent mixed convection
Westhoff, Andreas and Schmeling, Daniel and Bosbach, Johannes and Wagner, Claus (2010) Dynamics of large-scale structures and heat transfer in turbulent mixed convection. In: Proceedings of the Sixth Int. Symposium on turbulence, Heat and Mass Transfer Turbulence, Heat and Mass Tranfer 6. Begell House Inc. New York, Wallingford (UK). pp. 107-110. ISBN 978-1-56700-262-1.
Full text not available from this repository.
Official URL: http://www.ichmt.org
## Abstract
Low frequency oscillations have been observed in the heat transfer of mixed convection in a rectangular cavity with an aspect ratio of $\Gamma_{\mathrm{xz}} = 1$ and $\Gamma_{\mathrm{yz}} = 5$.~Mixed convective flow at $Ra = 2.4 \times 10^8$, $Re = 1.0 \times 10^{4}$, $Ar = 3.3$ and $Pr \approx 0.7$ has been studied to determine the nature of these oscillations. Therefore Particle Image Velocimetry (PIV) and temperature measurements have been performed under ambient and high pressure conditions. The PIV results have been analysed using Proper Orthogonal Decomposion (POD) to identify the characteristic frequencies of the coherent large-scale structures and their dynamics have been compared with the low frequency oscillations found in the heat transfer.
Document Type:Contribution to a Collection
Title:Dynamics of large-scale structures and heat transfer in turbulent mixed convection
Authors:
AuthorsInstitution or Email of Authors
Westhoff, Andreasandreas.westhoff@dlr.de
Schmeling, Danieldaniel.schmeling@dlr.de
Bosbach, Johannesjohannes.bosbach@dlr.de
Wagner, Clausclaus.wagner@dlr.de
Date:September 2010
Journal or Publication Title:Proceedings of the Sixth Int. Symposium on turbulence, Heat and Mass Transfer
Refereed publication:Yes
In ISI Web of Science:No
Page Range:pp. 107-110
Editors:
EditorsEmail
Hanjalic, K.UNSPECIFIED
Nagano, Y.UNSPECIFIED
Jakirlic, S.UNSPECIFIED
Publisher:Begell House Inc. New York, Wallingford (UK)
Series Name:Turbulence, Heat and Mass Tranfer 6
ISBN:978-1-56700-262-1
Status:Published
Keywords:mixed convection, turbulence, heat transfer, large scale circulations
HGF - Research field:Aeronautics, Space and Transport (old)
HGF - Program:Aeronautics
HGF - Program Themes:Aircraft Research
DLR - Research area:Aeronautics
DLR - Program:L AR - Aircraft Research
DLR - Research theme (Project):L - Systems & Cabin
Location: Göttingen
Institutes and Institutions:Institute of Aerodynamics and Flow Technology > Fluid Systems
Deposited By: Ilka Micknaus
Deposited On:20 Dec 2010 15:13
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2015-07-04 19:07:11
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https://kb.osu.edu/dspace/handle/1811/10019
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# THE $X^{2}\Pi-X^{2}\Pi (1-0)$ TRANSITIONS OF $^{14}N^{16}O, ^{15}N^{16}O, ^{14}{N^{17}}O, ^{14}{N}^{18}O$ NEAR 5,3 $\mu$
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/10019
Files Size Format View
1977-ME-04.jpg 123.1Kb JPEG image
Title: THE $X^{2}\Pi-X^{2}\Pi (1-0)$ TRANSITIONS OF $^{14}N^{16}O, ^{15}N^{16}O, ^{14}{N^{17}}O, ^{14}{N}^{18}O$ NEAR 5,3 $\mu$ Creators: Amiot, C.; Bacis, R.; Guelachvili, G. Issue Date: 1977 Publisher: Ohio State University Abstract: The spectra of the $X^{2}\pi-X^{2}\pi$(1$-$0) transitions of the $^{14}N^{16}O, ^{15}N^{16}O, ^{14}N^{17}O, ^{14}N^{18}O$ molecules have been recorded with the molecular Fourier transform interferometer. The resolution $(2.5 \times 10^{-3}$ $cm^{-1})$ allows us to separate the $\Lambda$ doublets in the transitions involving the $\Omega = 1/2$ states. The molecular constants have been obtained by numerical diagonalization of the Hamiltonian matrix. The high wavenumber precision permits verification of the excellent agreement for the ratio of the different parameters by isotopic substitution. Effects of various small parameters will be discussed. Description: Author Institution: Laboratoire de Physique Mol\'{e}Aculaire et d'Optique Atmospherique; Laboratoire de Spectrom\'{e}trie Ionique et Moleculaire, Universit\'{e} de Lyon; Laboratoire d'Infrarouge, Universit\'{e} de Paris XI URI: http://hdl.handle.net/1811/10019 Other Identifiers: 1977-ME-4
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2018-03-17 23:47:04
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https://uwaterloo.ca/social-development-lab/research/current-studies
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# Current studies
## Temperament Over Time Study ([TOTS]; in collaboration with the University of Maryland)
Currently, we are working on the TOTS which is a collaborative longitudinal study of children’s social development funded by the National Institute of Health. In this study, a large group of infants were recruited at the University of Maryland and have now been followed regularly for 15 years. In the Social Development Lab, we focus on how these children interact with their peers at each age of assessment. Using detailed behavioural coding schemes, we record both the quantity and quality of children’s interactions during unstructured and structured tasks.
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2019-09-21 13:03:46
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https://coding-gym.org/challenges/triple-sum/
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# Triple Sum
See the original problem on HackerRank.
## Solutions
Consider this test case:
Visually, for each element of B we should mark the elements in A and C that are less than or equal to that element:
And, for 3:
Performing the operation on the second 3 is redundant, because triplets must be unique.
Along the way, we could multiply the count of A‘s marked elements with the count of B‘s marked elements. All the results will be then accumulated.
For B[0] (2) we have:
1 2 3 A = [1] (1 element) C = [1, 2] (2 elements) Partial: 1x2=2
For B[1] (3) we have:
1 2 3 A = [1] (1 element) C = [1, 2, 3] (3 elements) Partial: 1x3=3
Thus, the total is 2+3=5.
Clearly, solving the problem with a brute force approach is easy and $$O(N^2)$$.
We can do better.
First of all, since triplets should be unique, we can remove all the duplicates from the arrays. Generally, we do it by sorting the arrays first.
To solve this problem, sorting is the key step.
Having the array sorted and without duplicates, we can think about an efficient approach. After all, for each element of B we need to find how many elements are smaller or equal in A and C.
### Upper Bound
Given a sorted array and a value, we know how to efficiently find the span of elements smaller than such value: it's a job for binary search and in particular for upper bound.
upper bound finds the first element strictly greater than the wanted value in logarithmic time. In our languages, upper bound (or whatever called) is generally implemented by returning a position (or iterator, index, etc).
For example, assuming 0-based indexing, if the upper bound of an element a is at position 2 then 2 elements appear before a (remember the array has no duplicates).
Thinking in C++ and Python, upper_bound and bisect return the first position after the wanted element. Thus, for each element bi of the array b we use upper_bound (or bisect) to find the right end (exclusive - as in C++ convention) of the span of elements less than or equal to bi. The size of such a span is the number of elements that we look for at each step.
Let's dive into some coding. Here is a C++ solution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 auto input(int len) { vector res(len); copy_n(istream_iterator(cin), len, begin(res)); sort(begin(res), end(res)); res.erase(unique(begin(res), end(res)), end(res)); return res; } int main() { int la, lb, lc; cin >> la >> lb >> lc; auto a = input(la); auto b = input(lb); auto c = input(lc); unsigned long long count = 0ull; for (auto bi : b) { auto lba = upper_bound(begin(a), end(a), bi); auto lbc = upper_bound(begin(c), end(c), bi); unsigned long long cardA = distance(begin(a), lba); unsigned long long cardC = distance(begin(c), lbc); count += cardA * cardC; } cout << count; }
The magic happens when we calculate the distance from the beginning, because in C++ we get an iterator back from upper_bound.
Looking carefully, there is a pattern emerging from the previous snippet. It's a reduce (foldl, accumulate)!
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 auto input(int len) { vector res(len); copy_n(istream_iterator(cin), len, begin(res)); sort(begin(res), end(res)); res.erase(unique(begin(res), end(res)), end(res)); return res; } int main() { int la, lb, lc; cin >> la >> lb >> lc; auto a = input(la); auto b = input(lb); auto c = input(lc); cout << accumulate(begin(b), end(b), 0ull, [&](auto cnt, auto bi){ return cnt + distance(begin(a), upper_bound(begin(a), end(a), bi)) * distance(begin(c), upper_bound(begin(c), end(c), bi)); }); }
Note that 64bit ints are needed or we overflow (here we used unsigned long long because, for example, on Windows using just long is not enought since it is 32bit).
Here is a more succinct Python solution (just the core part):
1 2 3 def triplets(a, b, c): a, b, c = sorted(set(a)), sorted(set(b)), sorted(set(c)) return sum(bisect.bisect(a, x) * bisect.bisect(c, x) for x in b)
set automatically removes duplicates and sorted sorts the result. Finally, the results coming from the list comprehension are summed up.
This solution is $$O(N \cdot logN)$$.
#### Simple optimization
If we go back to our first example, we note very clearly that when we iterate on B[i+1] (e.g. 3) we carry all the elements found for B[i] (e.g. 2). What if we start upper bound from the results of the previous?
Here is the idea (just the main part because the rest is unchanged):
1 2 3 4 5 6 7 8 9 auto lba=begin(a), lbc=begin(c); // maintain some state for (auto bi : b) { lba = upper_bound(lba, end(a), bi); lbc = upper_bound(lbc, end(c), bi); unsigned long long cardA = distance(begin(a), lba); unsigned long long cardC = distance(begin(c), lbc); count += cardA * cardC; }
This solution is still $$O(N \cdot logN)$$, although we ammortized the constant factor of the core algorithm since the span of binary search is decreased at every iteration.
Clearly, the tradeoff is that now the algorithm is stateful. If we want to massively parallelize it, we should use one of the other versions.
However, this optimization opens the doors to a better solution whose core part is linear.
### Sort + simple loop
There is a better solution which visits elements only once, so it's linear. Credits to Andrea Battistello, Simone Carani, Elia Giacobazzi and Roberto Melis who proposed similar solutions at Coding Gym Modena in December 2018.
Conceptually, starting from the previous solution, just replace upper_bound with find_if - whose predicate is just element > bi.
In other words, we find the positions of first elements greater than bi in both a and c, and we keep track of such positions (iterators, indexes) to start from there afterwards.
This way, each element is visited only once and the core part is linear (overall is still $$O(N \cdot logN)$$).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 using namespace std; auto input(int len) { vector res(len); copy_n(istream_iterator(cin), len, begin(res)); sort(begin(res), end(res)); res.erase(unique(begin(res), end(res)), end(res)); return res; } int main() { int la, lb, lc; cin >> la >> lb >> lc; auto a = input(la); auto b = input(lb); auto c = input(lc); unsigned long long count = 0ull; auto lba = begin(a), lbc = begin(c); for (auto bi : b) { lba = find_if(lba, end(a), [=](auto v){ return v>bi; }); lbc = find_if(lbc, end(c), [=](auto v){ return v>bi; }); count += distance(begin(a), lba) * distance(begin(c), lbc); } cout << count; }
1 2 3 4 5 6 7 8 9 10 11 12 13 14 def triplets(a, b, c): a, b, c = map(lambda x: sorted(set(x)), (a, b, c)) i_p = 0 i_r = 0 count = 0 for q in b: while i_p < len(a) and a[i_p] <= q: i_p += 1 while i_r < len(c) and c[i_r] <= q: i_r += 1 count += i_p * i_r return count
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2020-01-21 10:29:10
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http://matematika.reseneulohy.cz/3172/attaining-a-maximum
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## Attaining a maximum
The function $$f\colon \space \mathbb R^2 \to \mathbb R$$ is defined as $f(x, y) = \frac{1}{x^2 + y^2 + (x\cos(y) - 2x - 3e^y)^2 + 2}.$
Prove that $$f$$ attains its maximum value on $$\mathbb R$$.
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2022-05-26 13:33:10
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https://blender.stackexchange.com/questions/5154/blender-not-exporting-ambient-coefficient-in-mtl?noredirect=1
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# Blender not exporting Ambient Coefficient in .MTL? [duplicate]
Here are my settings:
Export settings:
In '.mtl'
newmtl c0
Ns 19.607843
Ka 0.000000 0.000000 0.000000
Kd 0.000000 0.077046 0.638270
Ks 0.882353 0.882353 0.882353
Ni 1.000000
d 1.000000
illum 2
• I don't think that Ambient Cooefficient is supported by .mtl. There's more info on .mtl files here: web.archive.org/web/20080813073052/http://local.wasp.uwa.edu.au/… – CharlesL Nov 28 '13 at 18:51
• It would help if we knew what you were trying to do and what you expect the Ambient value to be. This Ambient value is related to the render engine and doesn't really have anything to do with the material as it could be used outside Blender. – Greg Zaal Nov 28 '13 at 18:51
• – iKlsR Nov 28 '13 at 18:55
• Basically you can see Specular which is 'white' is the value 'Ks', Diffuse is 'Kd', but Ambient which is 1.0 is just 0.0 in 'Ka'. How do I change the value of 'Ka' when my materials are exported? – Joshua Barnett Nov 28 '13 at 19:42
• @CharlesL: from the link you provided: "To specify the ambient reflectivity of the current material, you can use the "Ka" statement (...)" – Polosson Nov 29 '13 at 0:05
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2020-10-01 16:55:58
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http://kitchingroup.cheme.cmu.edu/blog/2017/09/25/New-publication-in-J-Phys-Chem-Lett/
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## New publication in J. Phys. Chem. Lett.
| categories: | tags: | View Comments
DFT calculations are extensively used to predict the chemical properties of metal alloy surfaces, but they are expensive which limits the number of calculations that can be practically be calculated. In this paper, we explore a perturbation approach known as alchemy to take previously calculated results and extend them to new compositions. We use oxygen reduction as a prototype reaction, and show that alchemy is often much faster than DFT, with an accuracy within 0.1 eV of the DFT. There are cases where the accuracy is not as good suggesting that further improvements to the perturbation model could be beneficial. Overall, alchemy appears to be a useful tool in high-throughput screening research.
@article{saravanan-2017-alchem-predic,
title = {Alchemical Predictions for Computational Catalysis: Potential
and Limitations},
year = 2017,
Author = {Saravanan, Karthikeyan and Kitchin, John R. and von
Lilienfeld, O. Anatole and Keith, John A.},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.7b01974},
Doi = {10.1021/acs.jpclett.7b01974},
Eprint = {https://doi.org/10.1021/acs.jpclett.7b01974},
Journal = {The Journal of Physical Chemistry Letters},
Note = {PMID: 28938798},
Number = {ja},
Pages = {null},
Url = {https://doi.org/10.1021/acs.jpclett.7b01974},
Volume = 0,
}
Copyright (C) 2017 by John Kitchin. See the License for information about copying.
org-mode source
Org-mode version = 9.0.7
blog comments powered by Disqus
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2018-05-23 20:19:58
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http://mathematica.stackexchange.com/questions/3081/copying-an-image-from-the-clipboard-modifying-it-and-returning-it-to-the-clipbo
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# Copying an image from the clipboard, modifying it and returning it to the clipboard
I have a Mathematica script to help me copy images from PDFs and then make the (presumably white or nearly white) background transparent. Typically, I am looking at the PDF in Adobe Acrobat and I use the snapshot tool to copy the image to Microsoft Window's clipboard. Then I paste the image into mathematica and assign it the name q. Where I typed {IMAGE} below is where I paste the image into the code:
q = {IMAGE};
p = ColorNegate[Binarize[ImageApply[Min, q], 0.99]];
s = SetAlphaChannel[q, p];
Export["img.png", s];
I run the code and I get a PNG with a transparent background and then I can insert the png with the transparent background into OneNote.
The point of doing this is images with a transparent background are easier to arrange since the don't obscure the other texts. This makes it a snap to place graphs and equations in my notes.
I would like to make this even easier by enhancing my script. This is why I am asking for help. My goals are:
1. Is there some Mathematica command I can use to copy the image from the clipboard automatically without having to manually paste it.
2. Is there a way to get Mathematica to place the image back into the clipboard with a transparent background, thus cutting out the step of using a file.
What I have found is if I use a command like CopyToClipboard[s], it doesn't work.
1. Are there any other enhancements that would make the whole process more automated?
(A side note: it's been a while since I wrote this code, but if I remember correctly, I convert the image to black and white, figure out the background then use the pixels that were designated as background to figure out which pixels to set as transparent in the original image.)
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See this question for how to make a white background transparent. – R. M. Mar 16 '12 at 13:49
@R.M:Thanks for the link. My main interest is copying the image from the clipboard and copying back the modified image to the clipboard. The part of my code that makes the background transparent actually works quite well for my needs. Thanks. – Henry B Mar 16 '12 at 13:55
@HenryB I think you should post a separate question about the clipboard. That is unrelated to image processing. – Szabolcs Mar 16 '12 at 14:05
@HenryB I suggest you change the title and the details of your question if you're not interested in the transparent part – R. M. Mar 16 '12 at 14:08
Thanks for the advice. – Henry B Mar 16 '12 at 15:11
Here are two functions that'll do what you need.
putClipboardImage[img_Image] := Module[{nb},
nb = CreateDocument[{}, Visible -> False, WindowSelected -> False];
NotebookWrite[nb,
Cell[BoxData@ToBoxes@Image[img, Magnification -> 1]]];
SelectionMove[nb, All, CellContents];
FrontEndTokenExecute[nb, "CopySpecial", "MGF"];
NotebookClose[nb]
]
getClipboardImage[] := Module[{tag},
Catch[NotebookGet@ClipboardNotebook[] /.
r_RasterBox :>
Block[{},
Throw[Image[First[r], "Byte", ColorSpace -> "RGB"], tag] /;
True];
\$Failed, tag]
]
Warning: if the image is very small, this method will pad it with some wide pixels. It is copying the cell contents, not the image itself. If the image is small than the cell height, it'll get padded.
### Copying
The reason the image can't be pasted to every program when using CopyToClipboard is that it is placed onto the clipboard as a metafile (as well as a Mathematica expression), but not as a bitmap.
My function works by writing the image into a new hidden notebook, then invoking the Edit -> Copy As -> Bitmap command programmatically, to ensure that the image is placed on the clipboard as a bitmap. This works on Windows, but on OS X it's probably necessary to change MGF to something else, as the same command is not available (try PDF). On Linux this functionality is simply not available.
### Pasting
Pasting works by accessing the special object ClipboardNotebook[] and reading its contents. The unusual looking ReplaceAll - Throw - Catch construct is just a performance optimization to avoid unpacking the array representing the image data.
I used the same techniques in the image uploader palette.
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Thanks for the excellent answer. One problem is the transparent background. It seems that when I paste the image back to the clipboard using your code, it has a non-transparent background again. Should I perhaps ask a separate question on this issue? – Henry B Mar 17 '12 at 12:41
At the end of my code, I export the img to a file ( Export["img.png", s]; ). Maybe a reference to the file can somehow be stored in the clipboard. Not sure how this works ie. if a reference to a file is stored in the clipboard and that's why it works to copy the img from my desktop and paste the file into OneNote. – Henry B Mar 17 '12 at 12:50
@Henry To my knowledge, it's not possible to store an image with alpha channel on the Windows clipboard---at least in the standard format. Applications-specific formats are fine but can't be pasted anywhere except the application that produced them (as you discovered). About the reference, I'm not sure how that would work, or how to do that. I don't know much about the clipboard either. – Szabolcs Mar 17 '12 at 17:25
@Henry After opening up Dito ditto-cp.sourceforge.net in windows it appears file references are stored using CF_DROP. On a side it appears transperency was discussed over at stackoverflow.com/questions/15689541/… and they managed to create a semi working solution. – William May 27 '13 at 20:22
@HenryB I wrote up some code that copies mathematica PNGs to the clipboard over here mathematica.stackexchange.com/questions/26105/… – William Jul 7 '13 at 4:01
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2015-10-09 14:06:34
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https://openstax.org/books/college-algebra-corequisite-support/pages/2-key-concepts
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College Algebra with Corequisite Support
# Key Concepts
College Algebra with Corequisite SupportKey Concepts
### 2.1The Rectangular Coordinate Systems and Graphs
• We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis. See Example 1.
• An equation can be graphed in the plane by creating a table of values and plotting points. See Example 2.
• Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y=_____. See Example 3.
• Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the axes. See Example 4.
• The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See Example 5 and Example 6.
• The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See Example 7 and Example 8.
### 2.2Linear Equations in One Variable
• We can solve linear equations in one variable in the form $ax+b=0 ax+b=0$ using standard algebraic properties. See Example 1 and Example 2.
• A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See Example 3 and Example 4.
• All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See Example 5 and Example 6 and Example 7.
• Given two points, we can find the slope of a line using the slope formula. See Example 8.
• We can identify the slope and y-intercept of an equation in slope-intercept form. See Example 9.
• We can find the equation of a line given the slope and a point. See Example 10.
• We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See Example 11.
• The standard form of a line has no fractions. See Example 12.
• Horizontal lines have a slope of zero and are defined as $y=c, y=c,$ where c is a constant.
• Vertical lines have an undefined slope (zero in the denominator), and are defined as $x=c, x=c,$ where c is a constant. See Example 13.
• Parallel lines have the same slope and different y-intercepts. See Example 14 and Example 15.
• Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See Example 16.
### 2.3Models and Applications
• A linear equation can be used to solve for an unknown in a number problem. See Example 1.
• Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See Example 2.
• There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the $d=rt d=rt$ formula. See Example 3.
• Many geometry problems are solved using the perimeter formula $P=2L+2W, P=2L+2W,$ the area formula $A=LW, A=LW,$ or the volume formula $V=LWH. V=LWH.$ See Example 4, Example 5, and Example 6.
### 2.4Complex Numbers
• The square root of any negative number can be written as a multiple of $i. i.$ See Example 1.
• To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See Example 2.
• Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See Example 3.
• Complex numbers can be multiplied and divided.
• To multiply complex numbers, distribute just as with polynomials. See Example 4 and Example 5.
• To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See Example 6 and Example 7.
• The powers of $i i$ are cyclic, repeating every fourth one. See Example 8.
• Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-product property is then used to find solutions. See Example 1, Example 2, and Example 3.
• Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See Example 4 and Example 5.
• Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See Example 6 and Example 7.
• Completing the square is a method of solving quadratic equations when the equation cannot be factored. See Example 8.
• A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See Example 9 and Example 10.
• The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See Example 11.
• The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See Example 12.
### 2.6Other Types of Equations
• Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See Example 1, Example 2, and Example 3.
• Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See Example 4 and Example 5.
• We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See Example 6 and Example 7.
• To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See Example 8.
• Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See Example 9 and Example 10.
• Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See Example 11.
### 2.7Linear Inequalities and Absolute Value Inequalities
• Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See Table 1 and Example 2.
• Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See Example 3, Example 4, Example 5, and Example 6.
• Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See Example 7 and Example 8.
• Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See Example 9 and Example 10.
• Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See Example 11.
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2023-01-30 00:40:41
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https://socratic.org/questions/589263057c01490182c4c9d6
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# Question #4c9d6
Feb 2, 2017
Titanium dioxide, titanium(IV) oxide.
#### Explanation:
You are dealing with an ionic compound that contains the cation of a transition metal, so right from the start, you should be aware that its name will include a Roman numeral that describes its oxidation state.
We use Roman numerals to describe the oxidation states of transition metals because these elements can have moe than one oxidation state.
Notice that one formula unit of this compound contains $2$ oxide anions, $2 \times {\text{O}}^{2 -}$.
This means that in order to balance the overall negative charge coming from the two oxide anions, the cation must carry a $4 +$ charge.
Therefore, you can say that this ionic compound contains titanium(IV) cations, ${\text{Ti}}^{4 +}$. The IV Roman numeral is used to show that the cation has a $+ 4$ oxidation state.
The name of the ionic compound will be
${\text{TiO}}_{2} \to$ titanium(IV) oxide
A more common name used for titanium(IV) oxide is
${\text{TiO}}_{2} \to$ titanium dioxide
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2022-01-18 20:08:48
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https://motls.blogspot.com/2009/09/kyoto-ii-obama-vs-eurocrats.html?m=1
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Wednesday, September 16, 2009
Kyoto II: Obama vs Eurocrats
An entertaining split between Europe and America has emerged concerning the question how the carbon emissions reductions should be achieved in individual nations.
Obama and Barroso in Prague, April 2009. Things may have been different then.
As The Telegraph, The Guardian, and everyone else reports, Europe and America differ in their opinion how the internal rules to reduce the CO2 production should be set.
The European politicians think that Kyoto I has been such an amazing success ;-) that it should be repeated and its successes should be amplified. Among other things, it means that all nations should adopt the same internal mechanisms to punish the CO2 emissions. The U.S. economy should be controlled by the Eurocrats in Brussels in the same way as any other decent EU country and Barack Obama should remain what he is appreciated for, namely a puppet of the global political correctness headquarters that should stay in Brussels.
On the other hand, Barack Obama himself dared to disagree. Kyoto I hasn't been a sufficiently huge disaster so the U.S. president wants to engineer an even better scheme. As the first post-Hoover protectionist president of a country that rejected Kyoto I and is going to reject Kyoto II as long as it is isomorphic (and gives a free pass to the poorer emerging markets), he thinks that every country should be allowed to decide about its own methods to achieve the targets and the carbon flows in America should remain uncontrollable by the EU and the U.N. That's quite a heresy for the EU, comrade Obama! ;-)
Even Steven Chu has warned that deep CO2 reductions cannot be achieved politically in the U.S. Why doesn't he follow the example of the tall and strong Napoleon in France who defeated 74% of the French citizens and imposed a carbon tax upon them? ;-) Sarkozy also wants to start a world trade war by a new CO2 border tax. Swedish EU presidency also urges the U.S. Senate to behave; if they won't, the U.S. Senators will be spanked just like any bad EU kids. ;-)
It's not hard to understand Europe's newly gained self-confidence with respect to America. The Made-In-America downturn has allowed Europe to surpass North America as the wealthiest region of the world. And the future fate of the U.S. dollar (now at 1.475 per euro, or 17 crowns per dollar) - whose reserve status is being questioned by all members of BRIC as well as others (everyone can see that the U.S. may suffer from the same kind of an irresponsible socialist government as everyone else) - may turn out to have something to do with this picture.
The declared purpose of the December 2009 negotiations in Copenhagen that will hopefully fail completely is to save the Earth if not the multiverse. The UAH AMSU data see the average annual and global brightness temperature of the Earth to be close to minus 15.5 °C. Ban Ki-Moon and similar stellar scientists have calculated that if the temperature exceeds f***ing frying minus 13.5 °C which is by 2 °C higher, all of us are going to evaporate or transform into plasma and the Universe may decay into a different state, too. And I don't have to explain you the staggering statistical implications for the whole multiverse. ;-)
During the year, the brightness temperature oscillates approximately between -17 °C in January and -14 °C in July - because the variations of the landmass, which is mostly on the Northern Hemisphere, are more pronounced than the variations of the oceanic temperatures. The recent, 30-year trends indicate that the temperature is increasing roughly by 1 °C per century, so the catastrophic level when the temperature will oscillate between -15 °C and -12 °C could occur around the year 2200 or so - whether or not we will continue to use fossil fuels.
If you have ever experienced how much brutally hotter -12 °C is relatively to -14 °C, you must agree with all these guys that we're all doomed already next year - because we can already predict that the year 2200 will come - unless Obama and his compatriots will join the EU as obedient members. :-)
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2021-04-22 17:13:07
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http://mathoverflow.net/questions/83313/nonnegative-additive-functions-on-coherent-sheaves
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# Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well known to the expert but I couldn't find a reference: is $g$ a constant multiple of generic rank? If true, do you know of any reference for this?
Notes:
1. If $X=\mathrm{Spec}\:R$ is affine with $R$ an integral domain, then a proof can be found in Northcott-Reufel, Theorem 2, p. 303. There are also other proofs.
2. If $X$ is a projective variety over a field, I think I can prove it, but I don't know any reference for this case.
I have a feeling this question must have been answered in K-theory.
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Basically you ask for homomorphisms $G_0(X) \to \mathbb{R}_+$. – Martin Brandenburg Dec 13 '11 at 8:48
To Martin: no, he is asking for homomorphisms G0(X)→ℝ that are non-negative on classes of coherent sheaves. – Angelo Dec 13 '11 at 9:40
## 1 Answer
I suppose that "additive" means that "additive over short exact sequences". If so, this is does not seem too hard, at least if $X$ is separated.
By noetherian induction, you may assume that for all proper integral subscheme $Y$ of $X$, the restriction of $g$ to $Y$ is given by a multiple of the generic rank at $Y$. But every coherent sheaf with support on $Y$ can be given by a successive extension of coherent sheaves of $\mathcal O_Y$-modules; hence the restriction of $g$ to sheaves supported on $Y$ is given by a multiple of the length of the stalk at the generic point of $Y$. On the other hand, for each $n > 0$ denote by $Y_n$ the subscheme defined by the $n^{\mathrm th}$ power of the sheaf of ideals of $Y$; the length of the stalk of $\mathcal O_{Y_n}$ at the generic point of $Y$ is unbounded, but by the positivity of $g$ the value of $g(\mathcal O_{Y_n})$ is bounded by $g(\mathcal O_X)$. Hence this multiple is 0, and $g$ is 0 on all torsion sheaves. In particular, if $F \to G$ is a generic isomorphism of coherent sheaves, $g(F) = g(G)$.
But on the other hand if $F$ is a coherent sheaf of generic rank $r$, there exists homomorphisms $F \to G$ and $\mathcal O_X^r \to G$ of coherent sheaves that are generic isomorphisms; hence $g(F) = rg(\mathcal O_X)$. The conclusion follows.
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This is great, thank you! – Mahdi Majidi-Zolbanin Dec 13 '11 at 14:01
Perhaps in line 9, you meant "is bounded by $g(\mathcal O_X)$" instead of $\mathcal O_Y$. – Hailong Dao Dec 13 '11 at 16:32
To Hailong: yes, thank you. I edited the post. – Angelo Dec 13 '11 at 18:42
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2015-05-04 14:11:23
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http://math.stackexchange.com/tags/random-variables/hot
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# Tag Info
3
To find the p.d.f of the ratio $\frac{Y}{X+Y}$, let us first write its c.d.f. Since $X$ and $Y$ are always positive, their ratio is also positive and, therefore, for $0\leq t\lt1$ we can write: $P\left(\frac{Y}{X+Y}\leq t\right)=P\left(Y\leq \frac{t}{1-t}X\right)=\int_{0}^{\infty }\left(\int_{0}^{\frac{t}{1-t}x}f_{X}(x)f_{Y}(y)dy\right)dx$ as ...
2
For $x>1$, we have $$\mathbb P(\xi_n\leqslant x) = \mathbb P\left(\bigcap_{i=1}^n \left\{\eta_i\leqslant x\right\}\right)=\prod_{i=1}^n\mathbb P\left(\eta_i\leqslant x \right) = \left(1 - x^{-\alpha}\right)^n.$$ Hence \begin{align} \mathbb P(\zeta_n\leqslant x) &= \mathbb P\left(\xi_n n^{-\frac1\alpha}\leqslant x\right)\\ &= \mathbb ...
2
For any $p \geq 1$, we have $$|x+y|^p \leq 2^p (|x|^p+|y|^p),$$ and therefore \begin{align*} \mathbb{E}(|X_n-X|^p \mid \mathcal{F}) &\leq 2^p \mathbb{E}(|X_n|^p \mid \mathcal{F}) + 2^p \mathbb{E}(|Y|^p \mid \mathcal{F}) \\ &\leq 2^p \mathbb{E}(|X|^p \mid \mathcal{F}) + 2^p \mathbb{E}(|Y|^p \mid \mathcal{F}). \end{align*} This shows that ...
2
Hint: $X^2< \frac12\iff |X|<\frac1{\sqrt2}$. If $X$ and $Y$ are independent, then $|X|$ and $|Y|$ are also independent. Complete the sentence: if $A$ and $B$ are independent, then $P(A, B) = ...$
2
Since $X$ and $Y$ are independent, we have $$\Pr[(X^2 \le 1/2) \cap (|Y| \le 1/2)] = \Pr[X^2 \le 1/2]\Pr[Y \le 1/2].$$ Since they are uniform on $[-1,1]$, we then have $$\Pr[X^2 \le 1/2] = \Pr[-1/\sqrt{2} \le X \le 1/\sqrt{2}] = \frac{1/\sqrt{2} - (-1/\sqrt{2})}{1 - (-1)} = \frac{1}{\sqrt{2}},$$ and $$\Pr[|Y| \le 1/2] = \Pr[-1/2 \le Y \le 1/2] = ... 1 Notice that because of independence: $$P\left(X^2 < \frac{1}{2}, |Y| < \frac{1}{2} \right) = P\left(X^2 < \frac{1}{2}\right) P\left(|Y| < \frac{1}{2} \right)$$ Analyzing the X term: P\left(X^2 < \frac{1}{2}\right) = P \left( -\frac{1}{\sqrt{2}} < X < \frac{1}{\sqrt{2}} \right) = ... 1 The answer to your first question is yes: if c \ne 0 and c \in \mathbb R, then cX \sim \operatorname{Normal}(c\mu, |c|\sigma) if X is normal with mean \mu and standard deviation \sigma. This is because the normal distribution belongs to a location-scale family: its PDF is$$f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/(2\sigma^2)}, \quad ...
1
The probability of extinction is the smallest positive root of $$G_O(z)=z$$ Where $O$ denotes the offspring distribution, and $G_O(z)$ its generating function at $z$. It is easily seen that $G_O(0)$ is the probability of extinction in the first generation. Second, if you know about generating functions, then you know that the sum: , where $X$ is ...
1
For any positive integer $n$, we have $T_n-S_n = g(T_n,S_n)$ where $g:\mathbb R^2\to \mathbb R$ is the map $(x,y)\mapsto x-y$. Given $t\in\mathbb R$, it is clear that $$g^{-1}((-\infty,t]) = \{(x,y):x-y\leqslant t\}$$ is a Lebesgue-measurable set in $\mathbb R^2$, and so $g$ is a measurable function. Since $\sigma(g(T_n,S_n))\subset \sigma(T_n,S_n)$, ...
1
Let $X, Y$ be random variables in $(\Omega, \mathcal B, \mathbb P)$. If $A \in \mathcal B$, then $1_A$ and $1_{A^C}$ are random variables. Note that $$Z = X1_A + Y1_{A^C}$$ Since sums or products of random variables in $(\Omega, \mathcal B, \mathbb P)$ are random variables in $(\Omega, \mathcal B, \mathbb P)$, $Z$ is a random variable in $(\Omega, ... 1 'if' Let$A_n^c := \{X_n > M\}$. By BCL1, we have $$P(\limsup A_n^C) = 0$$ $$\to P(\liminf A_n) = 1$$ $$\to \lim P(A_n) = 1$$ $$\to P(\bigcap_{n=1}^{\infty} A_n) = 1$$ $$\to \prod_{n=1}^{\infty} P(A_n) = 1 \ \text{Why?}$$ $$\to \forall n \in \mathbb N, P(A_n) = 1$$ $$\to \forall n \in \mathbb N, P\{X_n \le M\} = 1$$ $$\to P( \sup_{n \ge 1} (X_n) ... 1 No, there is nothing to be said about the Cesàro averages of the whole sequence. They can be as bad as the sequence itself. Indeed, given any weakly convergent sequence \{f_n\}, we can consider another sequence \{g_n\} defined as$$f_1,f_1, f_2,f_2,f_2,f_2, f_2,f_2,f_2,f_2, f_2,f_2,f_2,f_2, f_2,f_2,f_2,f_2, f_3, \dots $$where the term f_n appears ... 1$$E[(a-\bar{a})(a+\bar{a})]\leq E[|a-\bar{a}|(a+\bar{a})]\leq 2E[|a-\bar{a}|],$$where the second inequality follows from the fact that (a+\bar{a})\leq 2. 1 As 0 \leq a, \bar{a} \leq 1, we have$$(a-\bar{a})(a + \bar{a}) \leq |(a-\bar{a})(a+\bar{a})| = |a-\bar{a}| \cdot \underbrace{|a+\bar{a}|}_{= a+\bar{a} \leq 2}.$$Taking expectation on both sides, proves the inequality. 1 Assuming that$(\sup\limits_n Y_n)(x)=\sup\limits_{n}\{Y_n(x)\}$then indeed, there is nothing guaranteeing that the first random variable$Y_1$should be finite a.s and therefore there is no reason that your claim should be true in general. The example you posited disproves it, but you need to fix values for all the later$Y_n$as well (i.e. with$n\geq2$). ... 1 I was thinking say$m = 2$and$Y_1 = \infty$, then$\sup_{n \ge m} Y_n < \infty$, but$\sup_{n \ge 1} Y_n = \infty ?$You almost answered you own question. You need further assumption for$Y_n$with$n\geq 2$though. Say,$Y_n\equiv 1$for each$n$. Then the$Y_n\$'s are independent. (Why?) Then you can check that the quoted statement is not true.
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-11-27 14:10:10
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http://www.heldermann.de/JCA/JCA24/JCA241/jca24023.htm
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Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 24 (2017), No. 1, 333--347 Copyright Heldermann Verlag 2017 A Note on the Extension of Continuous Convex Functions from Subspaces Carlo Alberto De Bernardi Dip. di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy carloalberto.debernardi@gmail.com [Abstract-pdf] Let $Y$ be a subspace of a real normed space $X$. We say that the couple $(X,Y)$ has the {\em $\mathrm{CE}$-property} (convex extension property'') if each continuous convex function on $Y$ admits a continuous convex extension defined on $X$. By using techniques of Johnson and Zippin, we prove the following results about the $\mathrm{CE}$-property: if $X$ is the $c_0(\Gamma)$-sum or the $\ell_p(\Gamma)$-sum (\$1 Keywords: Convex function, extension, normed linear space. MSC: 52A41; 26B25, 46A99 [ Fulltext-pdf (179 KB)] for subscribers only.
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2018-11-14 01:26:49
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http://flippedcoin.info/reference/find_congruent_models.html
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Find congruent models to a simple binomial model This will find the parameter values for other models that equal the likelihood for a simple binomial model. This may not be the MLE for these other models
find_congruent_models(
)
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2023-03-30 15:02:06
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https://quanteda.io/reference/textstat_frequency.html
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Produces counts and document frequencies summaries of the features in a dfm, optionally grouped by a docvars variable or other supplied grouping variable.
textstat_frequency(
x,
n = NULL,
groups = NULL,
ties_method = c("min", "average", "first", "random", "max", "dense"),
...
)
Arguments
x a dfm object (optional) integer specifying the top n features to be returned, within group if groups is specified either: a character vector containing the names of document variables to be used for grouping; or a factor or object that can be coerced into a factor equal in length or rows to the number of documents. NA values of the grouping value are dropped. See groups for details. character string specifying how ties are treated. See data.table::frank() for details. Unlike that function, however, the default is "min", so that frequencies of 10, 10, 11 would be ranked 1, 1, 3. additional arguments passed to dfm_group(). This can be useful in passing force = TRUE, for instance, if you are grouping a dfm that has been weighted.
Value
a data.frame containing the following variables:
feature
(character) the feature
frequency
count of the feature
rank
rank of the feature, where 1 indicates the greatest frequency
docfreq
document frequency of the feature, as a count (the number of documents in which this feature occurred at least once)
docfreq
document frequency of the feature, as a count
group
(only if groups is specified) the label of the group. If the features have been grouped, then all counts, ranks, and document frequencies are within group. If groups is not specified, the group column is omitted from the returned data.frame.
textstat_frequency returns a data.frame of features and their term and document frequencies within groups.
Examples
set.seed(20)
dfmat1 <- dfm(c("a a b b c d", "a d d d", "a a a"))
textstat_frequency(dfmat1)#> feature frequency rank docfreq group
#> 1 a 6 1 3 all
#> 2 d 4 2 2 all
#> 3 b 2 3 1 all
#> 4 c 1 4 1 alltextstat_frequency(dfmat1, groups = c("one", "two", "one"), ties_method = "first")#> feature frequency rank docfreq group
#> 1 a 5 1 2 one
#> 2 b 2 2 1 one
#> 3 c 1 3 1 one
#> 4 d 1 4 1 one
#> 5 d 3 1 1 two
#> 6 a 1 2 1 twotextstat_frequency(dfmat1, groups = c("one", "two", "one"), ties_method = "dense")#> feature frequency rank docfreq group
#> 1 a 5 1 2 one
#> 2 b 2 2 1 one
#> 3 c 1 3 1 one
#> 4 d 1 3 1 one
#> 5 d 3 1 1 two
#> 6 a 1 2 1 two
dfmat2 <- corpus_subset(data_corpus_inaugural, President == "Obama") %>%
dfm(remove_punct = TRUE, remove = stopwords("english"))
tstat1 <- textstat_frequency(dfmat2)
head(tstat1, 10)#> feature frequency rank docfreq group
#> 1 us 44 1 2 all
#> 2 must 25 2 2 all
#> 3 can 20 3 2 all
#> 4 nation 18 4 2 all
#> 5 people 18 4 2 all
#> 6 new 17 6 2 all
#> 7 time 16 7 2 all
#> 8 every 15 8 2 all
#> 9 america 14 9 2 all
#> 10 now 11 10 2 all
# \donttest{
# plot 20 most frequent words
library("ggplot2")
ggplot(tstat1[1:20, ], aes(x = reorder(feature, frequency), y = frequency)) +
geom_point() +
coord_flip() +
labs(x = NULL, y = "Frequency")
# plot relative frequencies by group
dfmat3 <- data_corpus_inaugural %>%
corpus_subset(Year > 2000) %>%
dfm(remove = stopwords("english"), remove_punct = TRUE) %>%
dfm_group(groups = "President") %>%
dfm_weight(scheme = "prop")
# calculate relative frequency by president
tstat2 <- textstat_frequency(dfmat3, n = 10, groups = "President")
# plot frequencies
ggplot(data = tstat2, aes(x = factor(nrow(tstat2):1), y = frequency)) +
geom_point() +
facet_wrap(~ group, scales = "free") +
coord_flip() +
scale_x_discrete(breaks = nrow(tstat2):1,
labels = tstat2\$feature) +
labs(x = NULL, y = "Relative frequency")# }
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2020-09-19 18:12:51
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https://math.stackexchange.com/questions/870282/what-does-the-colon-use-in-the-decimal-place-mean
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# What does the colon use in the decimal place mean?
When I type into wolfram the query, exp(1) in base 100, the answer comes back "2.71:82:81...."
What does the colon (:) mean?
• I would guess that the base-100 digits are the pairs separated by colons. For example, the first base-100 digit past the radix point is 71. – MPW Jul 17 '14 at 21:31
• to expand on what MPW said, you'd need 100 different characters to not need colons – Jam Jul 17 '14 at 21:33
You are asking for a number to be expressed in base $100$. That means the first "place" after the "decimal" point is in units of $1/100$, the second "place" after the "decimal" point is in units of $(1/100)^2$, and so on. Each "place" is populated by a whole number ("digit") from 0 to 99. Because we do not have distinct symbols for each of those possible digits, each one of those base-100 "digits" is actually written using a pair of base-10 digits. This makes reading the number hard, so it is helpful to use a delimiter to keep the place values separated for readability's sake.
The particular number you are looking at is $$2 + 71(1/100) + 82(1/100)^2 + 81(1/100)^3 + \dots$$
• When we write 1000 in base 3, we write it as $1101001$. We interpret this as $1 \cdot 3^6 + 1 \cdot 3^5 +1 \cdot 3^3 + 1$, but that is actually a base 10 expression -- which you can tell because it contains not just the symbol 6, but also a 5 and a 3, none of which are legal symbols in base 3. If we wanted to write the same expanded form using only base 3 symbols, we would write it as $1 \cdot 10^{20} + 1 \cdot 10^{12} + 1 \cdot 10^{10} + 1$. – mweiss Jul 18 '14 at 23:13
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2021-07-27 17:39:47
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https://deap.readthedocs.io/en/latest/porting.html
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# Porting Guide¶
DEAP development’s high velocity and our refusal to be at the mercy of backward compatibility can sometime induce minor headaches to our users. This concise guide should help you port your code from the latest version minus 0.1 to the current version.
## General¶
1. The algorithms from the algorithms module now return a tuple of 2-elements : the population and a Logbook.
2. Replace every call to DTM by calls to SCOOP.
3. Statistics and logging of data are accomplished by two distinct objects: Statistics and Logbook. Read the tutorial on logging statistics.
4. Replace EvolutionLogger by Logbook.
5. Replace usage of tools.mean(), tools.var(), tools.std(), and tools.median() by their Numpy equivalent.
6. If the fitness has multiple objectives, add the keyword argument axis=0 when registering statistical function.
## Genetic Algorithms (GA)¶
1. Replace every call to the function cxTwoPoints() by a call to cxTwoPoint().
2. Remove any import of cTools. If you need a faster implementation of the non-dominated sort, use sortLogNondominated().
3. When inheriting from Numpy, you must manually copy the slices and compare individuals with numpy comparators. See the Inheriting from Numpy tutorial.
## Genetic Programming (GP)¶
1. Specify a name as the first argument of every call to addEphemeralConstant().
2. Replace every call to lambdify() and evaluate() by a call to compile().
3. Remove the pset attribute from every create() call when creating a primitive tree class.
4. In the toolbox, register the primitive set as the pset argument of the following mutation operator: mutUniform(), mutNodeReplacement() and mutInsert().
5. Replace every call to the function genRamped() by a call to genHalfAndHalf().
6. Replace every call to stringify() by a call to str() or remove the call completely.
7. Replace every call to lambdifyADF() by a call to compileADF().
8. Replace the decorators staticDepthLimit() and staticSizeLimit() by staticLimit(). To specify a limit on either depth, size or any other attribute, it is now required to specify a key function. See staticLimit() documentation for more information.
### Strongly Typed Genetic Programming (STGP)¶
1. PrimitiveSetTyped method now requires type arguments to be defined as classes instead of string, for example float instead of "float".
## Evolution Strategy (ES)¶
1. Replace every call to the function cxESTwoPoints() by a call to cxESTwoPoint().
## Still having problem?¶
We have overlooked something and your code is still not working? No problem, contact us on the deap users list at http://groups.google.com/group/deap-users and we will get you out of trouble in no time.
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2021-01-28 14:28:52
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https://groupprops.subwiki.org/wiki/CA_not_implies_nilpotent
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# CA not implies nilpotent
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., CA-group) need not satisfy the second group property (i.e., nilpotent group)
View a complete list of group property non-implications | View a complete list of group property implications
## Statement
It is possible to have a CA-group $G$ that is not a nilpotent group. Here, CA means that the centralizer of every non-identity element is abelian. Equivalently, it means that every proper subgroup is either a centerless group or an abelian group.
This also shows that:
## Proof
### Finite example
Further information: symmetric group:S3, subgroup structure of symmetric group:S3
The group symmetric group:S3, defined as the group of permutations on $\{ 1,2,3 \}$, is a CA-group that is nontrivial and centerless, and therefore not nilpotent. To see that it is CA, note that it is centerless, and all its proper subgroups are abelian.
### Infinite example
Further information: infinite dihedral group
Consider the infinite dihedral group:
$G := \langle a,x \mid x = x^{-1}, xax^{-1} = a^{-1} \rangle$
This is centerless, and therefore, not nilpotent. On the other hand, the centralizer of every non-identity element is abelian. To see this, note that:
• The centralizer of any non-identity element inside $\langle a \rangle$ is precisely $\langle a \rangle$. This is abelian. In fact, it is isomorphic to the group of integers.
• For any element outside $\langle a \rangle$, the element has order two and its centralizer is the cyclic subgroup it generates. This is abelian, and is isomorphic to cyclic group:Z2.
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2020-02-28 13:15:55
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https://pointatinfinityblog.wordpress.com/category/art/
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# Life on the Poincaré Disk
Just at this time I left Caen, where I was then living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake I verified the result at my leisure.
-Henri Poincaré, Science and Method
You’re out for a walk one day, contemplating the world, and you suddenly have an out-of-body experience, your perspective floating high above your corporeal self. As you rise, everything seems perfectly normal at first, but, when you reach a sufficient altitude, you notice something strange: your body appears to be at the center of a perfect circle, beyond which there is simply…nothing!
You watch yourself walk towards the edge of the circle. It initially looks like you will reach the edge in a surprisingly short amount of time, but, as you continue watching, you notice yourself getting smaller and slowing down. By the time you are halfway to the edge, you are moving at only 3/4 of your original speed. When you are 3/4 of the way to the edge, you are moving at only 7/16 of your original speed. Maybe you will never reach the edge after all? What is happening?
At some point, you see your physical self notice some friends, standing some distance away in the circle. You wave to one another, and your friends beckon you over. You start walking toward them, but, strangely, you walk in what looks not to be a straight line but rather an arc, curving in towards the center of the circle before curving outward again to meet your friends. And, equally curiously, your friends don’t appear to be surprised or annoyed by your seemingly inefficient route. You puzzle things over for a few seconds before having a moment of insight. ‘Oh!’ you think. ‘My physical body is living on a Poincaré disk model for hyperbolic geometry, which my mind has somehow transcended during this out-of-body experience. Of course!”
The Poincaré disk model, which was actually put forth by Eugenio Beltrami, is one of the first and, to my mind, most elegant models of non-Euclidean geometry. Recall from our previous post that a Euclidean geometry is a geometry satisfying Euclid’s five postulates. The first four of these postulates are simple and self-evident. The fifth, known as the Parallel Postulate (recall also that two lines are parallel if they do not intersect), is unsatisfyingly complex and non-immediate. To refresh our memories, here is an equivalent form of the Parallel Postulate, known as Playfair’s Axiom:
Given any line $\ell$ and any point $P$ not on $\ell$, there is exactly one line through $P$ that is parallel to $\ell$.
A non-Euclidean geometry is a geometry that satisfies the first four postulates of Euclid but fails to satisfy the Parallel Postulate. Non-Euclidean geometries began to be seriously investigated in the 19th century; Beltrami, working in the context of Euclidean geometry, was the first to actually produce models of non-Euclidean geometry, thus proving that, supposing Euclidean geometry is consistent, then so is non-Euclidean geometry.
The Poincaré disk model, one of Beltrami’s models, is a model for hyperbolic geometry, in which the Parallel Postulate is replaced by the following statement:
Given any line $\ell$ and any point $P$ not on $\ell$, there are at least two distinct lines through $P$ that are parallel to $\ell$.
Points and lines are the basic objects of geometry, so, to describe the Poincaré disk model, we must first describe the set of points and lines of the model. The set of points of the model is the set of points strictly inside a given circle. For concreteness, let us suppose we are working on the Cartesian plane, and let us take the unit circle, i.e., the circle of radius one, centered at the origin, as our given circle. The points in the Poincaré disk model are then the points in the plane whose distances from the origin are strictly less than one.
Lines in the Poincaré disk model (which we will sometimes call hyperbolic lines) are arcs formed by taking one of the following type of objects and intersecting it with the unit disk:
1. Straight lines (in the Euclidean sense) through the center of the circle.
2. Circles (in the Euclidean sense) that are perpendicular to the unit circle.
(These can, of course, be seen as two instances of the same thing, if one takes the viewpoint that, in Euclidean space, straight lines are just circles of infinite radius.)
It’s already pretty easy to see that this geometry satisfies our hyperbolic replacement of the Parallel Postulate. In fact, given a line $\ell$ and a point $P$ not on $\ell,$ there are infinitely many lines through $P$ parallel to $\ell$. Here’s an illustration of a typical case, with three parallel lines drawn:
We’re not quite able right now to prove that the disk model satisfies the first four of Euclid’s postulates, in part because we haven’t yet specified what it means for two line segments in the model to be be congruent (we don’t, for example, have a notion of distance in our model yet). We’ll get to this in just a minute, but let us first show that our model satisfies the first postulate: Given any two distinct points, there is a line containing both of them.
To this end, let $A$ and $B$ be two points in the disk. If the (Euclidean) line that contains $A$ and $B$ passes through the center of the disk, then this is also a line in the disk model, and we are done. Otherwise, the (Euclidean) line that contains $A$ and $B$ does not pass through the center of the disk. In this case, we use the magic of circle inversion, which we saw in a previous post. Let $A'$ by the result of inverting $A$ across the unit circle. Now $A$, $A'$, and $B$ are distinct points in the Cartesian plane, so there is a unique circle (call it $\gamma$) containing all three. Since $A$ and $A'$ are both on the circle, it is perpendicular to the unit circle. Therefore, its intersection with the unit disk is a line in the disk model containing both $A$ and $B.$ Here’s a picture:
We turn now to distance in the Poincaré disk model. And here, for the sake of brevity, I’m not even going to try to explain why things are they way they are but will just give you a formula. Given two points $A$ and $B$ in the disk, consider the hyperbolic line containing them, and let $P$ and $Q$ be the points where this line meets the boundary circle (with $P$ closer to $A$ and $Q$ closer to $B$). Then the hyperbolic distance between $A$ and $B$ is given by:
$d(A,B) = \mathrm{ln}(\frac{|PB|\cdot|AQ|}{|PA|\cdot|BQ|})$.
This is likely inscrutable right now. That’s fine. Let’s think about what it means for this to be the correct notion of distance, though. For one thing, it means that, given two points in the disk model, the shortest path between them is not, in general, the straight Euclidean line that connects them, but rather the hyperbolic line that connects them. This explains your body’s behavior in the story at the start of this post. When you were walking over to your friends, what appeared to your mind (which was outside the disk, in the Euclidean realm) as a curved arc, and therefore an inefficient path, was in fact a hyperbolic line and, because your body was inside the hyperbolic disk, the shortest path between you and your friends.
This notion of distance also means that distances inside the disk which appear equal to an external Euclidean observer in fact get longer and longer the closer they are to the edge of the disk. This is also consistent with the observations at the beginning of the post: as your body got further toward the edge of the disk, it appeared from an external viewpoint to be moving more and more slowly. From a viewpoint inside the disk, though, it was moving at constant speed and would never reach the edge of the disk, which is infinitely far away. The disk appears bounded from the external Euclidean view, but from within it is entirely unbounded and limitless.
Let’s close by looking at two familiar shapes, interpreted in the hyperbolic disk. First, circles. Recall that a circle is simply the set of points that are some fixed distance away from a given center. Now, what happens when we interpret this definition inside the hyperbolic disk? Perhaps somewhat surprisingly, we get Euclidean circles! (Sort of.) To be more precise, hyperbolic circles in the Poincaré disk model are precisely the Euclidean circles that lie entirely within the disk. (I’m not going to go through the tedious calculations to prove this; I’ll leave that up to you…) Beware, though! The hyperbolic center of the circle is generally different from the Euclidean center. (This should make sense if you think about our distance definition. The hyperbolic center will be further toward the edge of the disk than the Euclidean center, coinciding only if the Euclidean center of the circle is in fact the center of the hyperbolic disk.)
Next, triangles. A triangle is, of course, a polygon with three sides. This definition works perfectly fine in hyperbolic geometry; we simply require that our sides are hyperbolic line segments rather than Euclidean line segments. If we assume the first four of Euclid’s postulates, then the Parallel Postulate is actually equivalent to the statement that the sum of the interior angles of a triangle is 180 degrees. In the Poincaré disk model (and, in fact, in any model of hyperbolic geometry) all triangles have angles that sum to less than 180 degrees. This should be evident if we look at a typical triangle:
Things become interesting when you start to ask how much less than 180 degrees a hyperbolic triangle has. The remarkable fact is that the number of degrees in a hyperbolic triangle is dependent entirely on its (hyperbolic) area! The smaller a triangle is, the larger the sum of its interior angles: as triangles get smaller and smaller, approaching a single point, the sum of their angles approaches 180 degrees from below. Correspondingly, as triangles get larger, the sum of their angles approaches 0 degrees. In fact if we consider an “ideal triangle”, in which the three vertices are in fact points on the bounding circle (and thus not real points in the disk model), then the sum of the angles of this “triangle” is actually 0 degrees!
A consequence of this is the fact that, in the Poincaré disk model, if two triangles are similar, then they are in fact congruent!
This leads us to our final topic: one of the perks of living in a Poincaré disk model. Perhaps the most frequent complaint I hear from people living on a Euclidean plane is that there aren’t enough ways to tile the plane with triangles. Countless people come up to me and say, “Chris, I want to tile the plane with triangles, and I want this tiling to have the following two pleasing properties:
1. All of the triangles are congruent, they don’t overlap, and they fill the entire plane.
2. At every vertex of the tiling, all angles meeting that vertex are the same.
But there are only four essentially different ways of doing this, and I’m tired of all of them! What should I do?”
(Exercise for the reader: Find all four such tilings!)
It just so happens that I have a simple answer for these people: “Move to a Poincaré disk model, where there are infinitely many tilings with these properties!” Here are just a few (all by Tamfang and in the public domain):
I’ll leave you with that! Hyperbolic geometry is fascinating, and I encourage you to investigate further on your own. The previous mentioned Euclid and Beyond, by Hartshorne, is a nice place to start.
This also wraps up (for now, at least) a couple of multi-part investigations here at Point at Infinity: a look at the interesting geometry of circles, which started in our post on circle inversion, and a look at various notions of independence in mathematics, the other posts being here and here. Join us next time for something new!
Cover Image: M. C. Escher, Circle Limit III
# Infinite Acceleration: Risset Rhythms
In our most recent post, we took a look at and a listen to Shepard tones and their cousins, Shepard-Risset glissandos, which are tones or sequences of tones that create the illusion of perpetually rising (or falling) pitch. The illusion is created by overlaying a number of tones, separated by octaves, rising in unison. The volumes gradually increase from low pitch to middle pitch and gradually decrease from middle pitch to high pitch, leading to a fairly seamless continuous tone.
The same idea can be applied, mutatis mutandis, to percussive loops instead of tones, and to speed instead of pitch, thus creating the illusion of a rhythmic track that is perpetually speeding up (or slowing down). (The mechanism is exactly the same as that of the Shepard tone, so rather than provide an explanation here, I will simply refer the reader to the previous post.) Such a rhythm is known as a Risset rhythm.
I coded up some very basic examples on Supercollider. Here’s an accelerating Risset rhythm:
And a decelerating Risset rhythm:
Here’s a more complex Risset rhythm:
And, finally, a piece of electronic music employing Risset rhythms: “Calculus,” by Stretta.
# Infinite Ascent: Shepard Tones
Have you ever been watching a movie and noticed that the musical score was seeming, impossibly, to be perpetually rising, ratcheting up the intensity of the film more and more? Or perhaps it seemed to be perpetually falling, creating a deeper and deeper sense of doom onscreen? If so, it is likely that this effect was achieved using a Shepard tone, a way of simulating an unbounded auditory ascent (or descent) in a bounded range.
To understand how Shepard tones work, let’s look at a simplified implementation of one. We will have three musical voices (middle, low, and high), with an octave between successive voices. The voices then start to move, in unison, and always an octave apart, up through a single octave, over, say, five seconds. As they go, though, they also change their volumes: the middle voice stays at full volume the whole time, the low voice gradually increases from zero volume to full volume, and the high voice gradually decreases from full volume to zero volume. The result will simply sound like a tone rising through an octave, and it can be represented visually as follows.
This by itself is nothing special, though. The trick of the Shepard tone is that this pattern is then repeated over, and over, and over again. Each repetition of the pattern sounds like a tone ascending an octave, but, because of the volume modulation, successive patterns are aurally glued together: the low voice from one cycle leads seamlessly to the middle voice of the next, the middle voice from one cycle leads seamlessly to the high voice of the next, and the high voice simply fades away. The result sounds like a perpetually increasing tone.
Note the similarity to the visual barber pole illusion, in which a rotating pole causes stripes to appear to be perpetually rising. Also, this whole story can be turned upside down, which will lead to a perpetually falling tone.
Let’s hear some Shepard tones in action! Now, in practice, using only three voices does not create a particularly convincing illusion, so, to make these sounds, I used nine voices, spread across nine octaves. Also, linearly varying the volume, as in the above visualization, seems to make it more noticeable when voices enter or fade away, so I used something more like a bell curve.
(Technical notes: These Shepard tones were created in Supercollider, using modified code written by Eli Fieldsteel, from whose YouTube tutorials I have learned a great deal of what I know about Supercollider. Also, I used a formant oscillator instead of the more traditional sine oscillator.)
First, a simple ascending Shepard tone:
The effect becomes more convincing, and the tone more interesting, if multiple Shepard tones are played simultaneously at a fixed interval. Here, we have two ascending Shepard tones separated by a tritone, a.k.a. the devil’s interval, a.k.a. half an octave:
Next, three descending Shepard tones, arranged in a minor triad:
Finally, two Shepard tones, with one ascending and the other descending:
The origins of the Shepard tone lie with Roger Shepard, a 20th-century American cognitive scientist, as a sequence of discrete notes. The continuous Shepard scale, or Shepard-Risset glissando, which our code approximates, was introduced by French composer Jean-Claude Risset, who perhaps most notably used it in his Computer Suite from Little Boy from 1968.
More recently, it has prominently been deployed by Christopher Nolan and Hans Zimmer, as the basis for the Batpod sound in The Dark Knight and in the Dunkirk soundtrack.
Cover image: M.C. Escher, Waterfall
# Surreal Numbers (Universal Structures III)
“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”
-John Horton Conway
# Dante, Einstein, and the Shape of the World
Last week, we began a series of posts dedicated to thinking about immortality. If we want to even pretend to think precisely about immortality, we will have to consider some fundamental questions. What does it mean to be immortal? What does it mean to live forever? Are these the same thing? And since immortality is inextricably tied up in one’s relationship with time, we must think about the nature of time itself. Is there a difference between external time and personal time? What is the shape of time? Is time linear? Circular? Finite? Infinite?
Of course, we exist not just across time but across space as well, so the same questions become relevant when asked about space. What is the shape of space? Is it finite? Infinite? It is not hard to see how this question would have a significant bearing on our thinking about immortality. In a finite universe (or, more precisely, a universe in which only finitely many different configurations of matter are possible), an immortal being would encounter the same situations over and over again, would think the same thoughts over and over again, would have the same conversations over and over again. Would such a life be desirable? (It is not clear that this repetition would be avoidable even in an infinite universe, but more on that later.)
Today, we are going to take a little historical detour to look at the shape of the universe, a trip that will take us from Ptolemy to Dante to Einstein, a trip that will uncover a remarkable confluence of poetry and physics.
One of the dominant cosmological views from ancient Greece and the Middle Ages was that of the Ptolemaic, or Aristotelian, universe. In this image of the world, Earth is the fixed, immobile center of the universe, surrounded by concentric, rotating spheres. The first seven of these spheres contain the seven “planets”: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. Surrounding these spheres is a sphere containing the fixed stars. This is the outermost sphere visible from Earth, but there is still another sphere outside it: the Primum Mobile, or “Prime Mover,” which gives motion to all of the spheres inside it. (In some accounts the Primum Mobile is itself divided into three concentric spheres: the Crystalline Heaven, the First Moveable, and the Empyrean. In some other accounts, the Empyrean (higher heaven, which, in the Christianity of the Middle Ages, became the realm of God and the angels) exists outside of the Primum Mobile.)
This account is naturally vulnerable to an obvious question, a question which, though not exactly in the context of Ptolemaic cosmology, occupied me as a child lying awake at night and was famously asked by Archytas of Tarentum, a Greek philosopher from the fifth century BC: If the universe has an edge (the edge of the outermost sphere, in the Ptolemaic account), then what lies beyond that edge? One could of course assert that the Empyrean exists as an infinite space outside of the Primum Mobile, but this would run into two objections in the intellectual climate of both ancient Greece and Europe of the Middle Ages: it would compromise the aesthetically pleasing geometric image of the universe as a finite sequence of nested spheres, and it would go against a strong antipathy towards the infinite. Archytas’ question went largely unaddressed for almost two millennia, until Dante Alighieri, in the Divine Comedy, proposed a novel and prescient solution.
Before we dig into Dante, a quick mathematical lesson on generalized spheres. For a natural number $n$, an $n$-sphere is an $n$-dimensional manifold (i.e. a space which, at every point, locally looks like $n$-dimensional real Euclidean space) that is most easily represented, embedded in $n+1$-dimensional space, as the set of all points at some fixed positive distance (the “radius” of the sphere) from a given “center point.”
Perhaps some examples will clarify this definition. Let us consider, for various values of $n$, the $n$-sphere defined as the set of points in $(n+1)$-dimensional Euclidean space at distance 1 from the origin (i.e. the point (0,0,…,0)).
If $n=0$, this is the set of real numbers whose distance from 0 is equal to 1, which is simply two points: 1 and -1.
If $n=1$, this is just the set of points $(x,y)$ in the plane at a distance of 1 from $(0,0)$. This is the circle, centered at the origin, with radius 1.
If $n=2$, this is the set of points $(x,y,z)$ in 3-dimensional space at a distance 1 from the point $(0,0,0)$. This is the surface of a ball of radius 1, and is precisely the space typically conjured by the word “sphere.”
0-, 1-, and 2-spheres are all familiar objects; beyond this, we lose some ability to visualize $n$-spheres due to the difficulty of considering more than three spatial dimensions, but there are useful ways to think about higher-dimensional spheres by analogy with the more tangible lower-dimensional ones. Let us try to use these ideas to get some understanding of the 3-sphere.
First, note that, for a natural number $n$, the non-trivial “cross-sections” of an $n+1$-sphere are themselves $n$-spheres! For example, if a 1-sphere (i.e. circle) is intersected with a 1-dimensional Euclidean space (a line) in a non-trivial way, the result is a $0$-sphere (i.e. a pair of points). If a 2-sphere is intersected with a 2-dimensional Euclidean space (a plane) in a non-trivial way, the result is a 1-sphere (this is illustrated above in our picture of a 2-sphere). The same relationship holds for higher dimensional spheres: if a 3-sphere is intersected with a 3-dimensional Euclidean space in a non-trivial way, the result is a 2-sphere.
Suppose that you are a 2-dimensional person living in a 2-sphere universe. Let’s suppose, in fact, that you are living in the 2-sphere pictured above, with the 1-sphere “latitude lines” helpfully marked out for you. Let’s suppose that you begin at the “north pole” (i.e. the point at the top, in the center of the highest circle) and start moving in a fixed direction. At fixed intervals, you will encounter the 1-sphere latitude lines. For a while, these 1-spheres will be increasing in radius. This will make intuitive sense to you. You are moving “further out” in space; each successive circle “contains” the last and thus should be larger in radius. After you pass the “equator,” though, something curious starts happening. Even though you haven’t changed direction and still seem to be moving “further out,” the radii of the circles you encounter start shrinking. Eventually, you reach the “south pole.” You continue on your trip. The circles wax and wane in a now familiar way, and, finally, you return to where you started.
A similar story could be told about a 3-dimensional being exploring a 3-sphere. In fact, I think we could imagine this somewhat easily. Suppose that we in fact live in a 3-sphere. For illustration, let us place a “pole” of this 3-sphere at the center of the Earth. Now suppose that we, in some sort of tunnel-boring spaceship, begin at the center of the Earth and start moving in a fixed direction. For a while, we will encounter 2-sphere cross-sections of increasing radius. Of course, in the real world these are not explicitly marked (although, for a while, they can be nicely represented by the spherical layers of the Earth’s core and mantle, then the Earth’s surface, then the sphere marking the edge of the Earth’s atmosphere) but suppose that, in our imaginary world, someone has helpfully marked them. For a while, these successive 2-spheres have larger and larger radii, as is natural. Eventually, of course, they will start to shrink, contracting to a point before expanding and contracting as we return to our starting point at the Earth’s core.
Dante’s Divine Comedy, completed in 1320, is one of the great works of literature. In the first volume, Inferno, Dante is guided by Virgil through Hell, which exists inside the Earth, directly below Jerusalem (from where I happen to be writing this post). In the second volume, Purgatorio, Virgil leads Dante up Mount Purgatory, which is situated antipodally to Jerusalem and formed of the earth displaced by the creation of Hell. In the third volume, Paradiso, Dante swaps out Virgil for Beatrice and ascends from the peak of Mount Purgatory towards the heavens.
Dante’s conception of the universe is largely Ptolemaic, and most of Paradiso is spent traveling outward through the larger and larger spheres encircling the Earth. In Canto 28, Dante reaches the Primum Mobile and turns his attention outward to what lies beyond it. We are finally in a position to receive an answer to Archytas’ question, and the answer that Dante comes up with is surprising and elegant.
The structure of the Empyrean, which lies outside the Primum Mobile, is in large part a mirror image of the structure of the Ptolemaic universe, a revelation that is foreshadowed in the opening stanzas of the canto:
When she who makes my mind imparadised
Had told me of the truth that goes against
The present life of miserable mortals —
As someone who can notice in a mirror
A candle’s flame when it is lit behind him
Before he has a sight or thought of it,
And turns around to see if what the mirror
Tells him is true, and sees that it agrees
With it as notes are sung to music’s measure —
Even so I acted, as I well remember,
While gazing into the bright eyes of beauty
With which Love wove the cord to capture me.
When Dante looks into the Empyrean, he sees a sequence of concentric spheres, centered around an impossibly bright and dense point of light, expanding to meet him at the edge of the Primum Mobile:
I saw a Point that radiated light
So sharply that the eyelids which it flares on
Must close because of its intensity.
Whatever star looks smallest from the earth
Would look more like a moon if placed beside it,
As star is set next to another star.
Perhaps as close a halo seems to circle
The starlight radiance that paints it there
Around the thickest mists surrounding it,
As close a ring of fire spun about
The Point so fast that it would have outstripped
The motion orbiting the world most swiftly.
And this sphere was encircled by another,
That by a third, and the third by a fourth,
The fourth by a fifth, the fifth then by a sixth.
The seventh followed, by now spread so wide
That the whole arc of Juno’s messenger
Would be too narrow to encompass it.
So too the eighth and ninth, and each of them
Revolved more slowly in proportion to
The number of turns distant from the center.
This seemingly obscure final detail, that the spheres of the Empyrean spin increasingly slowly as they increase in size, and in distance from the point of light, turns out to be important. Dante is initially confused because, in the part of the Ptolemaic universe from the Earth out to the Primum Mobile, the spheres spin faster the larger they are; the fact that this is different in the Empyrean seems to break the nice symmetry he observes. Beatrice has a ready explanation, though: the overarching rule governing the speed at which the heavenly spheres rotate is not based on their size, but rather on their distance from God.
This is a telling explanation and seems to confirm that the picture Dante is painting of the universe is precisely that of a 3-sphere, with Satan, at the center of the Earth, at one pole and God, in the point of light, at the other. If Dante continues his outward journey from the edge of the Primum Mobile, he will pass through the spheres of the Empyrean in order of decreasing size, arriving finally at God. Note that this matches precisely the description given above of what it would be like to travel in a 3-sphere. Dante even helpfully provides a fourth dimension into which his 3-sphere universe is embedded: not a spatial dimension, but a dimension corresponding to speed of rotation!
(For completeness, let me mention that the spheres of the Empyrean are, in order of decreasing size and hence increasing proximity to God: Angels, Archangels, Principalities, Powers, Virtues, Dominions, Thrones, Cherubim, and Seraphim.)
Dante’s ingenious description of a finite universe helped the Church to argue against the existence of the infinite in the physical world. Throughout the Renaissance, Scientific Revolution, and Enlightenment, this position was gradually eroded in favor an increasingly accepted picture of infinite, flat space. A new surprise awaited, though, in the twentieth century.
In 1917, Einstein revolutionized cosmology with the introduction of general relativity, which provided an explanation of gravity as arising from geometric properties of space and time. Central to the theory are what are now known as the Einstein Field Equations, a system of equations that describes how gravity interacts with the curvature of space and time caused by the presence of mass and energy. In the 1920s, an exact solution to the field equations, under the assumptions that the universe is homogeneous and isotropic (roughly, has laws that are independent of absolute position and orientation, respectively), was isolated. This solution is known as the Friedmann-Lemaître-Robertson-Walker metric, after the four scientists who (independently) derived and analyzed the solution, and is given by the equation,
$ds^2 = -dt^2 + R^2(t)\left(\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2)\right)$,
where $k$ is a constant corresponding to the “curvature” of the universe. If $k = 0$, then the FLRW metric describes an infinite, “flat” Euclidean universe. If $k < 0$, then the metric describes an infinite, hyperbolic universe. If $k>0$, though, the metric describes a finite universe: a 3-sphere.
PS: Andrew Boorde, from whose book the above illustration of the Ptolemaic universe is taken, is a fascinating character. A young member of the Carthusian order, he was absolved from his vows in 1529, at the age of 39, as he was unable to adhere to the “rugorosite” of religion. He turned to medicine, and, in 1536, was sent by Thomas Cromwell on an expedition to determine foreign sentiment towards King Henry VIII. His travels took him throughout Europe and, eventually, to Jerusalem, and led to the writing of the Fyrst Boke of the Introduction of Knowledge, perhaps the earliest European guidebook. Also attributed to him (likely without merit) is Scoggin’s Jests, Full of Witty Mirth and Pleasant Shifts, Done by him in France and Other Places, Being a Preservative against Melancholy, a book which, along with Boord himself, plays a key role in Nicola Barker’s excellent novel, Darkmans.
Mark A. Peterson, “Dante and the 3-sphere,” American Journal of Physics, 1979.
Carlo Rovelli, “Some Considerations on Infinity in Physics,” and Anthony Aguirre, “Cosmological Intimations of Infinity,” both in Infinity: New Research Frontiers, edited by Michael Heller and W. Hugh Woodin.
Cover Image: Botticelli’s drawing of the Fixed Stars.
# The Infinite Art of Barcelona
I spent part of the last two weeks savoring the art and architecture of Barcelona, entranced by the work of two artistic giants of Catalonia: Joan Miró and Antoni Gaudí. Both artists seem to revel in dualities. They take the straight lines of the manmade world and bend them. They take the unruly forms of nature and impose upon them an order. They seek to contain the universe in a box, to express infinity in their necessarily finite works.
This is already too much writing from me. Simply enjoy the words and works of these two artists. (The Miró here is all from the Fundació Joan Miró and the Gaudí from the Sagrada Família.)
Look for the noise hidden in silence, the movement in immobility, life in inanimate things, the infinite in the finite, forms in a void, and myself in anonymity.
-Joan Miró
The straight line is the expression of the infinite.
-Antoni Gaudí
The straight line belongs to Man. The curved line belongs to God.
-Antoni Gaudí
# Infinite Picture Books
I’ve been traveling for the last month or so and thus have not been posting here. I have returned, though, so I will try to begin regularly updating again.
Today, just a short post to tell you to go have a look at some fantastic work by Richard Evan Schwartz of Brown University, who has two delightful (and occasionally terrifying) PDF picture books about infinity on his website:
Enjoy!
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2018-06-19 15:54:02
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http://math.nie.edu.sg/events/Colloquium/Ms.aspx
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Mathematics & Mathematics Education ACADEMIC GROUP
MME Staff and Graduate Student Colloquium 2019
Date: Monday 15 April 2019
Time: 4.30 pm – 8.30 pm
Venue: TR201 (Math Edn) & TR203 (Math)
Registration closes : 2 April 2019
## Mathematics Abstracts of Presentations
On an amazing identity of Ramanujan
Toh Pee Choon
In this talk, I will introduce and prove an amazing identity that involves the sum of two squares function. Although the identity is attributed to S. Ramanujan, it does not appear in any of his published works. I will also present generalizations of Ramanujan’s identity that were obtained jointly with H. H. Chan.
On $k$-restricted overpartitions
Uha
In this talk, we introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the 5-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo 5. Finally, we pose some problems for future work.
Scott’s Information system
Li Zhuolun
In his pedagogical effort to make domain theory more palatable to computer scientists and logicians, Dana Scott introduced the notion of information system. An information system is a formal way of setting up a universe A of tokens, an identified collection of subsets called Con, and an entailment relation between Con and A. This entailment relation is required to satisfy some very natural axioms, and it turns out that an information system induces a domain and vice versa. In this talk, this strong link between information system and domains is explained.
A short excursion into computational geometry
Elaine Wong
Computational geometry is a research field concerning both mathematics and computer science, devoted to designing, analysing and implementing algorithms that solve geometric problems. In this project, some fundamental problems arising in different application domains of computational geometry have been explored. The way that these problems can be transformed into purely geometric problems and how algorithms make use of these geometric properties to solve the problems have been studied. These problems include the closest pair problem, the convex hull and more.
Parking functions
Dong Fengming
The notion of a parking function was introduced by Konheim and Weiss in 1966. Suppose that there are $n$ drivers labeled $1,2,\cdots,n$ and $n$ parking spaces arranged in a line numbered $1,2,\cdots,n$. Assume that driver $i$ has its initial parking preference $f(i)$, where $1\le f(i)\le n$. Assume that these $n$ drivers enter the parking area in the order $1,2,\cdots,n$ and driver $i$ park at space $j$, where $j$ is the minimum number with $f(i)\le j\le n$ such that space $j$ is unoccupied by the previous drivers. If all drivers can park successfully by this rule, then $(f(1),f(2), \cdots,f(n))$ is called a {\it parking function} of length $n$. Mathematically, a function $f:N_n\rightarrow N_n$, where $N_n=\{1,2,\cdots,n\}$, is called {\it a parking function} if the inequality $|\{1\le i\le n: f(i)\le k\}|\ge k$ holds for each integer $k:1\le k\le n$. For example, for $n=2$, $(f(1),f(2))=(1,1)$, $(f(1),f(2))=(1,2)$ and $(f(1),f(2))=(2,1)$ are parking functions, but $(f(1),f(2))=(2,2)$ is not. It can be shown easily that $f:N_n\rightarrow N_n$ is a parking function if and only if there is a permutation $\pi_1,\pi_2,\cdots,\pi_n$ of $N_n$ such that $f(\pi_j)\le j$ holds for all $j=1,2,\cdots,n$. Konheim and Weiss proved that the number of parking functions of length $n$ is equal to $(n+1)^{n-1}$, which is equal to the number of spanning trees of the complete graph $K_{n+1}$.
Parking functions are related to many topics in combinatorial theory. In this talk, I will introduce various extensions of parking functions.
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2019-04-21 16:17:48
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https://math.stackexchange.com/questions/2534109/computing-type-ii-error-for-a-one-sided-normal-test
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# Computing Type II Error for a One-Sided Normal Test.
For a random sample $X_1,X_2,\ldots , X_{49}$ taken from a population having standard deviation 4, sample mean is found to be 20.3. Test the claim that the mean of the population is not less than 21 at 1% level of significance.Find the probability of Type II error if the population mean is 19.1
So to test for $H_0: \mu\geq 21$ versus alternate hypothesis $H_1: \mu < 21$ we calculate the test statistic $Z=\frac{20.3-21}{\frac{4}{\sqrt{49}}}=-1.225$ and since $-z_{0.01}=-2.326<-1.225$, we cannot reject the null hypothesis at 1% level of significance.
I am confused how to do the second part. I know that the Type 2 error will be $P(Accept \ H_0|\mu=19.1)$. How do I do this?
Intuitively, for the test you have $H_0: \mu \ge 21$ and $H_a: \mu < 21.$ From data you have $\bar X = 20.3,$ which is smaller then $\mu_0 = 21.$ However, the critical value for a test at level 1% is $c = 19.67.$ Because $\bar X > c,$ you find that $\bar X$ is not significantly smaller than $\mu_0.$
Computation using R: Under $H_0$ we have $\bar X \sim \mathsf{Norm}(21, 4/7);\,P(\bar X \le 19.671) = .01.$
qnorm(.01, 21, 4/7) # 'qnorm' is normal quantile function (inverse CDF)
## 19.67066 # 1% critical value
pnorm(19.671, 21, 4/7) # 'pnorm' is normal CDF
## 0.01001595 # verified
Now you wonder, whether a specific alternative value $\mu_a = 19.1 < 21$ might have yielded a value of $\bar X$ small enough to lead to rejection.
The Answer from @spaceisdarkgreen (+1) has done the power computation by standardizing, so that probabilities can be read from printed normal tables.
If we leave the problem on the original measurement scale, the following figure illustrates the situation. The blue curve (at right) is the hypothetical normal distribution of $\bar X \sim \mathsf{Norm}(\mu_0 = 21, \sigma = 2/7).$ The 1% significance level is the area under this curve to the left of the vertical line.
The orange curve is the alternative normal distribution of $\bar X \sim \mathsf{Norm}(\mu_a =19.1, \sigma = 2/7).$ The area to the left of the vertical line under this curve represents the power against alternative $H_a: \mu = \mu_a,$ which is $0.840.$ [The power is $1 - P(\text{Type II Error}).$]
Computation: Under $H_a: \mu_a = 19.1,$ we have $\bar X \sim \mathsf{Norm}(19.1, 4/7).$
pnorm(19.671, 19.1, 4/7)
## 0.8411632 # power against alternative 19.1
1 - pnorm(19.671, 19.1, 4/7)
## 0.1588368 # Type II error probability
Note: Some statistical calculators can be used to find the same normal probabilities I have found using R statistical software.
Addendum: Some textbooks reduce the computations shown by @spaceisdarkgreen to the following formula for Type II error of a one-sided test at level $\alpha$ against an alternative $\mu_a:$
$$\beta(\mu_a) = P\left(Z \le z_\alpha - \frac{|\mu_0-\mu_a|}{\sigma/\sqrt{n}} \right).$$ In your case this is $P(Z \le 2.326 - 3.325 = -0.999) = \Phi(-0.999) = 0.1589.$
Ref.: The displayed formula is copied from Sect 5.4 of Ott & Longnecker: Intro. to Statistical Methods and Data Analysis.
For the $1\%$ significance level, you have calculated you should reject if $$\frac{\sqrt{49}(21-\bar X)}{4} > 2.326,$$ so you need to calculate the probability that $$\frac{\sqrt{49}(21-\bar X)}{4} < 2.326$$ when $X_i\sim N(19.1,16).$
Under this assumption, you know that $$\frac{\sqrt{49}(19.1-\bar X)}{4} \sim N(0,1)$$ so you just need to rearrange: $$P\left(\frac{\sqrt{49}(21-\bar X)}{4} < 2.326\right)=P\left(\frac{\sqrt{49}(19.1-\bar X)}{4}+\frac{\sqrt{49}(1.9)}{4} < 2.326\right)\\=P\left(\frac{\sqrt{49}(19.1-\bar X)}{4}<-0.999\right) =\Phi(-0.999) = 15.9\%$$
Another way of saying this is that at effect size $\mu = 19.1$ and sample size $n=49,$ the test has $1-15.9\% = 84.1\%$ power.
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2020-10-22 21:44:18
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http://math.stackexchange.com/questions/133266/an-exercise-in-allen-hatchers-book-on-spectral-sequences
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# An Exercise in Allen Hatcher's book on Spectral Sequences
anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows:
For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ show that the associated action of $\pi_1K(C,1)=C$ on $H_*(K(A,1);G)$ is trivial if $A$, regarded as a subgroup of $B$, lies in the center of $B$.
Any help will be greatly appreciated.
-
The action of $C$ on $K(A,1)$ in the case when $A$ is abelian is given by applying $K(-, 1)$ to the conjugation maps $b: A\to A$, $a\mapsto bab^{-1}$. These maps are trivial on $A$ if $A$ is abelian. Of course, all these maps are trivial if $A$ is contained in the center of $B$. I don't know how you are defining the action of $C$ on $H_*(A, M)$ so I don't know how to prove these are the same. – Justin Young Apr 18 '12 at 18:54
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2015-05-23 01:35:12
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https://www.esaral.com/q/using-binomial-theorem-determine-which-number-is-larger-1-2-56322
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# Using binomial theorem determine which number is larger (1.2)
Question:
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Solution:
We have:
$(1.2)^{4000}=(1+0.2)^{4000}$
$={ }^{4000} C_{0}+{ }^{4000} C_{1} \times(0.2)^{1}+{ }^{4000} C_{2} \times(0.2)^{2}+\ldots{ }^{4000} C_{4000} \times(0.2)^{4000}$
$=1+4000 \times 0.2+$ other positive terms
$=1+800+$ other positive terms
$=801+$ other positive terms
$\because 801>800$
Hence, (1.2)4000 is greater than 800
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2023-03-27 03:03:02
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https://vulcanhammer.net/2018/06/06/more-uses-for-p-q-diagrams/
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Posted in Soil Mechanics
# More Uses for p-q Diagrams
In our last post on p-q diagrams we discussed their basic concept and application. In this post we’ll expand on that for two applications: using it to estimate the friction angle and cohesion for multiple triaxial tests, and using it to plot the failure function.
## Processing Triaxial Test Results
The process of determining internal friction angle and cohesion from successive triaxial tests (i.e., those where the confining stress is successively increased) is well known. In the case of two tests, using the standard $\sigma - \tau$ diagram, the tangent line between the two circles is unique (well, there are two of them, but the slopes and intercepts have opposite signs) as shown below.
If we use the p-q diagram, as we saw earlier, the process is even simpler, as two points have a unique line between them.
But what happens with three tests? Mathematically there is no guarantee of a unique line, and given the nature of geotechnical testing it is the extraordinary lab which could hit such as result. It’s also possible that the failure envelope is non-linear, as shown below.
So is there a way to at least get a decent approximation without guesswork or graphics skills? The answer is “yes” and it involves using p-q diagrams in conjunction with a spreadsheet. The mathematical concept behind this is here and we have an example to show how it is done. The problem is taken from Tchebotarioff’s (1951) classic soil mechanics test. The results of three triaxial tests are as follows, the failure stresses are in tsf:
Test $\sigma_3$ $\sigma_1$ p q 1 0.2 0.82 0.51 0.31 2 0.4 1.6 1 0.6 3 0.6 2.44 1.52 0.92
We’ve taken the liberty of computing the p and q values for each test. Now we can plot these in our spreadsheet.
We’ve also taken the liberty to use the spreadsheet’s “trend line” feature to plot a linear “curve fit” for the points. The slope of the equation $m=tan \delta = sin \phi = 0.6041$, which yields both $\delta = 31.1^\circ$ and $\phi = 37.2^\circ$. For the intercept $b = c\sqrt {1-\left (\tan(\delta)\right )^{2}} = 0.0001$, which means we can solve for the cohesion, but in this case the quantity is so small it’s probably best to assume that the cohesion is zero.
The $R^2$ value for this problem is very high, so the correlation is good. We can use this parameter to determine whether we have a good correlation or not. We can also use least-squares trend line analysis for non-linear failure envelopes, although when we consider the “kink” caused by preconsolidation this may not be as meaningful as one would like.
## Plotting the Failure Function
As mentioned earlier, the Mohr-Coulomb failure function is define in this way:
A little math transforms this into
$f=2\,q-2\,c\sqrt {1-\left (\tan(\delta)\right )^{2}}-2\,p\tan(\delta)$
or
$f=2\,q-2\,c\sqrt {1-\left (\sin(\phi)\right )^{2}}-2\,p\sin(\phi)$
Since $\delta$ and $c$ are known, this suggests that we can plot the failure function three-dimensionally. Consider the case where $\delta = \frac{\pi}{8}$ and $c = 5$. The p-q diagram for the failure envelope $f = 0$ is shown below.
If we plot the failure function three-dimensionally, we obtain this result:
The failure envelope of the previous diagram is the contour line which stops at the q-axis at around q = 4.6. Values below this line are negative and values above it are positive. Positive values of $f$ indicate failure and an illegal stress state. The failure function is used extensively in finite element analyses like this one.
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2018-06-20 01:39:04
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http://tex.stackexchange.com/tags/png/new
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# Tag Info
3
The test file test.tex for the examples below is the file from the question: \documentclass{article} \usepackage[margin=0cm,paperheight=20cm,paperwidth=5.5cm]{geometry} \begin{document} \noindent hello world \end{document} The following options of dvipng work for me: dvipng -D300 -O 1in,1in -T 5.5cm,20cm test Result: test1.png identify test1.png ...
1
To strip all .png files with ImageMagick, you can simply run the following command find . -type f -name "*.png" -exec convert {} -strip {} \;
5
Step 1 Make sure you have installed ImageMagick. Step 2 For the sake of simplicity, I create converter.tex which embed your code inside as follows, % this file name is converter.tex % compile it with pdflatex -shell-escape converter.tex \documentclass[preview,border=12pt]{standalone} \usepackage{filecontents} \begin{filecontents*}{diagram.tex} ...
4
Not exactly an answer to the question, but it is fun. \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{backgrounds} \begin{document} \begin{tikzpicture} \draw[yellow,yslant=0.5] (0,0) rectangle (2,2); \node[yslant=0.5,anchor=center,opacity=0.3] at (1,1.5) {\includegraphics[width=2cm,height=2cm]{example-image-a}}; ...
3
There are several questions on the site which address filling a shape with an image in TikZ, for example: TikZ: Drawing regular hexagons with pictures inside How to clip an external image inside an ellipse We can adapt those solutions to your 3D shape. We use \clip to restrict to the front square of the cube, then add the image, and finally we draw the ...
Top 50 recent answers are included
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2014-04-18 23:58:39
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https://mersenneforum.org/showthread.php?s=0be8a9c8a7515f97e1a0653f610cd50f&t=23963&page=4
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mersenneforum.org New Mersenne Software For Test Mersenne Prime Numbers On Android
User Name Remember Me? Password
Register FAQ Search Today's Posts Mark Forums Read
View Poll Results: What about This app you like it? Yes 4 36.36% No 3 27.27% Regular 1 9.09% Very bad 3 27.27% Voters: 11. You may not vote on this poll
2019-01-04, 02:41 #34 axn Jun 2003 47·109 Posts How difficult will it be to compile MLucas for android ? I would love to see the performance of an actual state-of-the-art code running on one of these.
2019-01-04, 06:53 #35 LaurV Romulan Interpreter Jun 2011 Thailand 100110000101112 Posts @OP: can you give us some info about the multiplication algorithm used? Is it school grade, is it karatsuba? FFT? As per firejuggler's suggestion, we tried to run this toy in an android emulator - our kingdom is Cortex M, but we have some android emus laying around, as we have colleagues who develop applications with them, for our customers. We wanted to see how the tests scale in time - if the application is bogus, like reading exponents from a list and saying prime/composite, and doing other things meantime, then the time-scaling would be also odd. We have some idea and could guess the multiplication algorithm used, if any tuning was done or it is just blind school-grade high-level stuff. And we consider this to be a good test, because it would not be easy for the guy to fake the process (which involves exponentiations and multiplications related not only to the size of the exponents, but also to the number of bits which are 1 involved). And our toys can also debug/disassemble code. But unluckily, we can not run it. Our emulator does not like it, and our colleagues who could help didn't come back from their NY holidays yet... Our advice for now: avoid it. P.S. Carlos, you seem to be the only one liking it? Can you detail why? Or was that a voting mistake? Last fiddled with by LaurV on 2019-01-04 at 06:58
2019-01-04, 07:11 #36 pinhodecarlos "Carlos Pinho" Oct 2011 Milton Keynes, UK 2×2,477 Posts Being Portuguese I’m being supportive with our Spanish brother but also I’m affected by flu since new year on my leave. Call me crazy: am I tired of fights with Spanish or is flu damaging my brain?!??? Or both?!
2019-01-04, 07:22 #37 LaurV Romulan Interpreter Jun 2011 Thailand 72·199 Posts Ok, that makes sense! (being supportive).
2019-01-04, 09:29 #38 GP2 Sep 2003 5·11·47 Posts Here is a Mersenne prime test. I didn't write it. It uses the GMP library. It prints a 64-bit hexadecimal residue: "0" for a Mersenne prime, non-zero otherwise. I compiled on Linux and ran on a single Skylake core on AWS. For 86243 it takes 11 seconds; for 216091 it takes 1 minute 38.5 seconds. And yet this is much slower than mprime/Prime95. Code: /* Adapted from http://rosettacode.org/wiki/Lucas-Lehmer_test#GMP */ #include #include #include void lucas_lehmer(unsigned long p, mpz_t *V) { mpz_t mp, t; unsigned long k; mpz_init_set_ui(t, p); mpz_init(mp); mpz_setbit(mp, p); mpz_sub_ui(mp, mp, 1); mpz_init_set_ui(*V, 4); for (k = 3; k <= p; k++) { mpz_mul(*V, *V, *V); mpz_sub_ui(*V, *V, 2); /* mpz_mod(*V, *V, mp) but more efficiently done given mod 2^p-1 */ if (mpz_sgn(*V) < 0) mpz_add(*V, *V, mp); /* while (n > mp) { n = (n >> p) + (n & mp) } if (n==mp) n=0 */ /* but in this case we can have at most one loop plus a carry */ mpz_tdiv_r_2exp(t, *V, p); mpz_tdiv_q_2exp(*V, *V, p); mpz_add(*V, *V, t); while (mpz_cmp(*V, mp) >= 0) mpz_sub(*V, *V, mp); } mpz_clear(t); mpz_clear(mp); /* Residue is returned in V */ } int main(int argc, char* argv[]) { mpz_t m64, V; /* We want to print a 64-bit residue */ mpz_init(m64); mpz_setbit(m64, 64); mpz_sub_ui(m64, m64, 1); unsigned long p; if (argc >= 2) { p = strtoul(argv[1], 0, 10); } else { fprintf(stderr, "Provide one argument, a Mersenne exponent, for instance: 11213\n"); return 1; } lucas_lehmer(p, &V); mpz_and(V, V, m64); mpz_out_str(stdout, 16, V); fprintf(stdout, "\n"); mpz_clear(m64); mpz_clear(V); return 0; } Last fiddled with by GP2 on 2019-01-04 at 09:38
2019-01-04, 09:48 #39
LaurV
Romulan Interpreter
Jun 2011
Thailand
72·199 Posts
Quote:
Originally Posted by LaurV Ok, that makes sense! (being supportive).
Please don't do this to my posts! Now there are two of you, and I don't know who to blame!
(now I know why Xyzzy gave the knife to Batalov too! )
2019-01-04, 15:53 #40
Batalov
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5·1,907 Posts
Quote:
Originally Posted by thorken M86243 In 3 minutes 7 seconds on s4 mini gt i9195 M216091 in 23min 7 sec in s4 mini gt-i9195
Quote:
Originally Posted by GP2 Here is a Mersenne prime test. I didn't write it. It uses the GMP library. It prints a 64-bit hexadecimal residue: "0" for a Mersenne prime, non-zero otherwise. I compiled on Linux and ran on a single Skylake core on AWS. For 86243 it takes 11 seconds; for 216091 it takes 1 minute 38.5 seconds.
Great job, Gord!
2019-01-04, 16:28 #41 axn Jun 2003 47·109 Posts 3 min 7 sec / 11 sec = 17x 23min 7 sec / 1 min 38.5 sec = 14x OP's code has better scaling than GMP-based one !
2019-01-04, 17:25 #42
Mysticial
Sep 2016
23×43 Posts
Quote:
Originally Posted by axn 3 min 7 sec / 11 sec = 17x 23min 7 sec / 1 min 38.5 sec = 14x OP's code has better scaling than GMP-based one !
Granted, GMP isn’t a very high bar to beat. Their refusal to use floating point means they don’t use any fast algorithms until much larger sizes.
2019-01-04, 17:34 #43
science_man_88
"Forget I exist"
Jul 2009
Dumbassville
26×131 Posts
Quote:
Originally Posted by axn 3 min 7 sec / 11 sec = 17x 23min 7 sec / 1 min 38.5 sec = 14x OP's code has better scaling than GMP-based one !
by a factor of roughly 1/2< ratio of n's > if my math is correct, but it in theory takes until exponents of 2.9 to 3 million to surpass it.
2019-01-04, 19:21 #44 kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 127738 Posts Fixed overhead can give lower order empirical timing scaling from low p measurements and make an implementation look like it's using a better algorithm than it is. In other words, the order bends upward to higher power of p at higher exponent, as setup overhead gets diluted by longer runtimes for higher exponent. This can be seen in runtime scaling measurements and plots I made for cllucas, CUDALucas and gpuowl. The more setup overhead, the lower the apparent order at low p. OpenCL's compile on the fly guaranteed around 2 seconds of setup overhead at each fft length. CUDALucas scaling: https://www.mersenneforum.org/showpo...3&postcount=2; gpuowl V5 scaling: https://www.mersenneforum.org/showpo...&postcount=10; prime95 scaling: https://www.mersenneforum.org/showpo...78&postcount=2 The per-iteration time scaling of gpulucas, cllucas, CUDALucas, and gpuowl v1.9 are compared at https://www.mersenneforum.org/showpo...76&postcount=8 I'd expect grammar-school multiplication based primality testing to scale as ~p3. With sufficient overhead, it may scale as p2.43 at the low end, as in my ancient inefficient primitive algorithm code from the late 80s and early 90s does at low p. (There, inefficient TF is part of the run time, as is printout of every iteration's res64 and very frequent state saves for resumption as insurance against the vagaries of MS-DOS and no UPS.) Thorken's timings indicate scaling of p2.42, very close to that. It could be grammar school, or something better.
Similar Threads Thread Thread Starter Forum Replies Last Post JonathanM Information & Answers 25 2020-06-16 02:47 paulunderwood Miscellaneous Math 18 2017-01-26 20:33 primus Miscellaneous Math 1 2014-10-12 09:25 allasc Math 33 2011-05-20 22:48 jocelynl Math 8 2006-10-20 19:36
All times are UTC. The time now is 15:05.
Thu Sep 23 15:05:03 UTC 2021 up 62 days, 9:34, 0 users, load averages: 4.01, 3.55, 3.40
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2021-09-23 15:05:03
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http://ayotzinapasomostodos.com/lib/convergent_series.htm
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# convergent series
Convergent Series
An infinite series for which the sequence of partial sumsconverges. For example, the sequence of partial sums of the series0.9 + 0.09 + 0.009 + 0.0009 + ··· is 0.9, 0.99, 0.999, 0.9999, .... This sequence converges to 1, so the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is convergent.
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2019-10-19 09:36:25
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https://www.tutorialspoint.com/program-to-find-the-mid-point-of-a-line-in-cplusplus
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# Program to find the mid-point of a line in C++
C++Server Side ProgrammingProgramming
In this problem, we are given two points A and B, starting and ending point of a line. Our task is to create a program to find the mid-point of a line in C++.
Problem Description − Here, we have a line with starting and ending points A(x1, y1) and B(x2, y2). And we need to find the mid-point of the line.
Let’s take an example to understand the problem,
## Input
a(x1, y1) = (4, -5)
b(x2, y2) = (-2, 6)
## Output
(1, 0.5)
## Explanation
(x1 + x2)/2 = 4 - 2 / 2 = 1
(y1 + y2)/2 = -5 + 6 / 2 = 0.5
## Solution Approach
To solve the problem, a simple method is using the geometrical formula for the mid of a line. The formula is given by,
Mid = ( ((x1 + x2)/2), ((y1 + y2)/2) )
Program to illustrate the working of our solution,
## Example
Live Demo
#include<iostream>
using namespace std;
int main() {
float point[2][2] = {{-4, 5}, {-2, 6}};
float midX = (float)(( point[0][0] + point[1][0])/2);
float midY = (float)(( point[0][1] + point[1][1])/2);
cout<<"The mid-points are ("<<midX<<" , "<<midY<<")";
return 0;
}
## Output
The mid-points are (-3 , 5.5)
Updated on 15-Sep-2020 15:24:12
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2022-08-13 02:35:52
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http://mathoverflow.net/feeds/question/64461
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Is there an elegant algebraic proof of this formula for quadratic field discriminants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:53:51Z http://mathoverflow.net/feeds/question/64461 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64461/is-there-an-elegant-algebraic-proof-of-this-formula-for-quadratic-field-discrimin Is there an elegant algebraic proof of this formula for quadratic field discriminants? Frank Thorne 2011-05-10T05:00:53Z 2011-05-13T17:59:54Z <p>Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "basically" $\zeta(s)/\zeta(2s)$. However, there is the funny business at 2, and one derives the formula</p> <p>$\sum D^{-s} = \frac{1}{2} \big( 4^{-s} - 1 \big) \frac{\zeta(s)}{\zeta(2s)} + \frac{1}{2} \big(1 - 4^{-s} \big) \frac{L(s, \chi_4)}{L(2s, \chi_4)},$</p> <p>which is a wee bit messy. (This formula, and all the subsequent ones, include 1 as a "quadratic field discriminant" for convenience.)</p> <p>However, I was reading a <a href="http://www.math.okstate.edu/~wrightd/Works/MyScans/TwistsOfIwasawaTate.pdf" rel="nofollow">fantastic paper by David Wright</a>, where he considers positive and negative discriminants together, in which case you have the much nicer formula</p> <p>$\sum |D|^{-s} = \big(1 - 2^{-s} + 2 \cdot 4^{-s} \big) \frac{\zeta(s)}{\zeta(2s)}.$</p> <p>This formula is easy to derive from scratch, but he derives it as a consequence of the beautiful formula</p> <p>$\sum |D|^{-s} = \prod_p \Big( \frac{1}{2} \sum_{[K_v : \mathbb{Q}_p] \leq 2} |\text{Disc}(K_v)|^s_p \Big).$</p> <p>He uses the parameterization of quadratic fields by $\mathbb{Q}^{\times} / (\mathbb{Q}^{\times})^2$, which may be thought of as $\text{GL}_1$-orbits on a one-dimensional prehomogeneous vector space, where $\text{GL}_1$ acts by $t(x) = t^2 x$ rather than the usual $t(x) = tx$. He then analyzes these orbits by means of an adelic zeta function; note that with the usual action you recover Tate's thesis.</p> <p>These formulas generalize quite a bit, with some complications, to $n$th-root extensions of any global field (with some restrictions on the characteristic). They also allow for twisting by characters, allowing (for example) a nice formula for $\sum \text{sgn}(D) |D|^{-s}.$ Note that the first formula considered looks nicer when viewed as a linear combination of $\sum |D|^{-s}$ and $\sum \text{sgn}(D) |D|^{-s}$.</p> <p>The MathSciNet review says that "similar results, however, can be obtained by class field theory", and this is also hinted at in Wright's paper, but the details aren't worked out. My first question is the following.</p> <p><strong>Is there an elegant algebraic proof of the above identity for $\sum |D|^{-s}$ and its generalizations?</strong></p> <p>And my second question, which is essentially a vaguer version of the first one, is:</p> <p><strong>What is the best way to think of these formulas?</strong></p> <p>I quite like the prehomogeneous vector space approach, but I imagine there might be a nice algebraic proof as well, and in particular some kind of local-to-global'' principle for quadratic discriminants. I am no expert in class field theory, and I am curious if any of these identities look simple and natural when viewed in the correct light.</p> <p>Thank you!</p> http://mathoverflow.net/questions/64461/is-there-an-elegant-algebraic-proof-of-this-formula-for-quadratic-field-discrimin/64592#64592 Answer by Franz Lemmermeyer for Is there an elegant algebraic proof of this formula for quadratic field discriminants? Franz Lemmermeyer 2011-05-11T10:50:19Z 2011-05-13T17:59:54Z <p>The discriminant $D$ of a quadratic number field can be written uniquely as a product of prime discriminants, namely $-4$, $\pm 8$, and $p^* = (-1)^{(p-1)/2}p$ for odd primes $p$. The Dirichlet series for odd discriminants therefore simply is $$\sum_{D \text{ odd}} |D|^{-s} = \prod_{p \text{ odd}} (1 + p^{-s}),$$ and the contribution of the even prime discriminants is taken care of by the factor $$1 + 4^{-s} + 2 \cdot 8^{-s}.$$ Both the beautiful as well as the much nicer formula now follow immediately.</p> <p>Factorization of quadratic discriminants into prime discriminants holds over totally real algebraic number fields with class number $1$ in the strict sense (see e.g. L. Goldstein, On prime discriminants, Nagoya Math. J. 45, 119-127 (1972); J. Sunley, Remarks concerning generalized prime discriminants, Boulder 1972; J. Sunley, Prime discriminants in real quadratic fields of narrow class number one, Carbondale 1979); a weaker version good enough for the purpose of counting discriminants works if the class number in the strict sense is odd.</p>
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2013-05-20 17:53:50
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https://www.zbmath.org/?q=an%3A1231.90088
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# zbMATH — the first resource for mathematics
Value of information in capacitated supply chains. (English) Zbl 1231.90088
Summary: We incorporate information flow between a supplier and a retailer in a two-echelon model that captures the capacitated setting of a typical supply chain. We consider three situations: (1) a traditional model where there is no information to the supplier prior to a demand to him except for past data; (2) the supplier knows the (s, S) policy used by the retailer as well as the end-item demand distribution; and (3) the supplier has full information about the state of the retailer. Order up-to policies continue to be optimal for models with information flow for the finite horizon, the infinite horizon discounted and the infinite horizon average cost cases. Study of these three models enables us to understand the relationships between capacity, inventory, and information at the supplier level, as well as how they are affected by the retailer’s (S - s) values and end-item demand distribution. We estimate the savings at the supplier due to information flow and study when information is most beneficial.
##### MSC:
90B06 Transportation, logistics and supply chain management
##### Keywords:
information sharing; $$(s, S)$$ policy
Full Text:
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2021-04-17 15:42:59
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http://cloudcomputing.sys-con.com/node/1094488
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Welcome!
Cloud Expo Authors:
Related Topics: Cloud Expo
Cloud Expo: Blog Post
## What is Elastic Computing?
For me, the economic-centric definition is the best way to apply elasticity to cloud computing
Recently there has been a lot of talk about private clouds, public clouds and inter clouds and the problem in attempting to define their key attributes. The problem facing all the various cloud terms is the apparent lack of quantifiable characteristics. In contrast most other areas of technology include standard characteristics such as units of measurement that allows us to define specific quantifiable aspects.
These aspects allow for the creation of complex models and formulas that enable a continued improvement and enhancement to the underlying platforms. I thought I'd briefly take a closer look at core concepts of elasticity and how it can be applied to cloud computing.
For a lot of companies looking at entering the cloud computing marketplace the lack of quantifiable capacity metrics has become a particular problem point. Currently trying to plan future resource & capacity requirements is an extremely difficult endeavor. Recent high profile cloud outages have made the need for quantifiable proactive capacity measurement painfully clear. When it comes to scaling infrastructure it seems that a lot of cloud computing has come down to a best guess mentality. (Although companies like SOASTA have done a great job at attempting to solve this problem) At the end of the day these outages are generally because of the unforeseen aspects, namely the inability to adequately determine and measure an application or infrastructure's breaking point. For this reason I believe that going forward the concept of elasticity will quickly become one of the most important aspects when designing most modern cloud centric system architectures.
To give you some background, I've been pitch my vision for elastic computing for almost 6 years. (I'm told others have also used the term before me as well) Generally my theory for Elastic Computing is in the ability to apply a quantifiable methodology that allows for the basis of an adaptive introspection with in a real time hybrid cloud centric infrastructure. But this doesn't actually address the question of "What is Elastic Computing" for that we need to dig deeper.
First we must look at what "elasticity" is and how it can be applied. According wikipedia, "in physics, elasticity is the physical property of a material when it deforms under stress (e.g. external forces), but returns to its original shape when the stress is removed. The relative amount of deformation is called the strain." Similarly elasticity applied to computing can be thought as the amount of strain an application or infrastructure can withstand while either expanding or contracting to meet the demands place on it.
One of the best examples of elasticity can be found in economics where elasticity is described as "the ratio of the percent change in one variable to the percent change in another variable. It is a tool for measuring the responsiveness of a function to changes in parameters in a relative way. A typical example is analysis of the elasticity of substitution, price and wealth. Popular in economics, elasticity is an approach used among empiricists because it is independent of units and thus simplifies data analysis."
For me, the economic-centric definition is the best way to apply elasticity to cloud computing. In this approach the main benefit is the ability to measure quantifiable capacity metrics using standard mathematical formulas such as "arc elasticity" -- a formula where the elasticity of one variable is measured with respect to another between two given points. The arc elasticity method of introspection is used when there is no general function for the relationship of two variables. (Think of an unknown or sudden increase in demand placed on an application or infrastructure) Therefore this formula provides the ability of to act as a predictive estimator for required system elasticity.
Wikipedia provides this economic example.
The y arc elasticity of x is defined as:
$E_{x,y} = \frac{\% \mbox{ change in } x}{\% \mbox{ change in } y}$
where the percentage change is calculated relative to the midpoint
$\% \mbox{ change in } x = \frac{x_2 - x_1}{(x_2 + x_1)/2}$
$\% \mbox{ change in } y = \frac{y_2 - y_1}{(y_2 + y_1)/2}$
What is Elastic Computing?
The quantifiable ability to manage, measure, predict and adapt responsiveness of an application based on real time demands placed on an infrastructure using a combination of local and remote computing resources.
## More Stories By Reuven Cohen
An instigator, part time provocateur, bootstrapper, amateur cloud lexicographer, and purveyor of random thoughts, 140 characters at a time.
Reuven is an early innovator in the cloud computing space as the founder of Enomaly in 2004 (Acquired by Virtustream in February 2012). Enomaly was among the first to develop a self service infrastructure as a service (IaaS) platform (ECP) circa 2005. As well as SpotCloud (2011) the first commodity style cloud computing Spot Market.
Reuven is also the co-creator of CloudCamp (100+ Cities around the Globe) CloudCamp is an unconference where early adopters of Cloud Computing technologies exchange ideas and is the largest of the ‘barcamp’ style of events.
Internet of Things and ‘The Physical Web’ By @ScottJenson | @ThingsExpo [#IoT]By Liz McMillanScott Jenson leads a project called The Physical Web within the Chrome team at Google. Project members are working to take the scalability and openness of the web and use it to talk to the exponentially exploding range of smart devices. Nearly every company today working on the IoT comes up with the same basic solution: use my server and you'll be fine. But if we really believe there will be trillions of these devices, that just can't scale. We need a system that is open a scalable and by using ...Jan. 31, 2015 02:00 AM EST Reads: 3,071 GE's Industrial Internet of Things as a Service | @ThingsExpo [#IoT]By Elizabeth WhiteIn their session at @ThingsExpo, Shyam Varan Nath, Principal Architect at GE, and Ibrahim Gokcen, who leads GE's advanced IoT analytics, focused on the Internet of Things / Industrial Internet and how to make it operational for business end-users. Learn about the challenges posed by machine and sensor data and how to marry it with enterprise data. They also discussed the tips and tricks to provide the Industrial Internet as an end-user consumable service using Big Data Analytics and Industrial C...Jan. 31, 2015 01:00 AM EST Reads: 2,941 CEO Power Panel on SYS-CON.tv | @CloudExpo [#Cloud]By Liz McMillanThings are being built upon cloud foundations to transform organizations. This CEO Power Panel at 15th Cloud Expo, moderated by Roger Strukhoff, Cloud Expo and @ThingsExpo conference chair, addressed the big issues involving these technologies and, more important, the results they will achieve. 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2015-01-31 07:02:58
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https://www.physicsforums.com/threads/a-limit-of-an-indexed-integral-at-closed-interval.649513/
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A limit of an indexed integral at closed interval
1. Nov 4, 2012
BaitiTamam
Let ρ(x) be a continuous function on ℝ, which evaluates ρ(x)=0 when |x|≥1 and that meets following.
∫[-1,1]ρ(x)dx=1
And let ψ(x) be a continuous function on interval [-1,1], prove
lim[n→∞] n∫[-1,1]ρ(nx)ψ(x)dx = ψ(0).
is denoted.
This is NOT a homework but a past exam problem of a college that has no answer.
I've been thinking for a whole day now.
What I have found so far:
First I defined sequencing functions as following:
F_n(x)=∫[-1,x]ρ(ny)ψ(y)dy
G_n(x)=∫[x,1]ρ(ny)ψ(y)dy
Now given a properly large natural number of n_0 which completes n_0≥1/y (where y≠0),
for all n∈N of n≥n_0, ρ(ny) evaluates 0. In this sense, taken a small positive number ε>0 instead of y, we only have to concern that of a small closed interval of [-ε,ε] as to nF_n(-ε) and nG_n(ε) gives 0 (for all n≥n_0≥1/ε).
So for all n same above,
n∫[-1,1]ρ(nx)ψ(x)dx = n∫[-ε,ε]ρ(nx)ψ(x)dx
And I have a stuck.
2. Nov 4, 2012
lurflurf
First change variables (just for simplicity)
n∫[-1,1]ρ(nx)ψ(x)dx =∫[-1,1]ρ(x)ψ(x/n)dx
now by the mean value theorem for integrals
∫[-1,1]ρ(x)ψ(x/n)dx =ψ(t/n)∫[-1,1]ρ(x)dx =ψ(t/n)
where |t|<1, but may depend upon ψ and n
consider
lim[n→∞] ψ(t/n)
3. Nov 5, 2012
BaitiTamam
>lurflurf
I got it. Changing variable was out of my thought.
Thank you:)
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2018-04-26 17:46:22
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https://danmackinlay.name/notebook/infinitesimal_generators.html
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# Infinitesimal generators
## Generators of the transition semi-group, connection to Kolmogorov forward equations
At first I found it hard to visualise infinitesimal generators but perhaps this simple diagram will help
This note exists because no one explained to me satisfactorily to me why I should care about infinitesimal generators. These mysterious creatures pop up in the study of certain continuous time Markov processes, such as stochastic differential equations driven by Lévy noise. To learn: connection to Koopman operators.
Infinitesimal generators have a simple natural interpretation in terms of evolution of the laws of Markov chains. TBD: going from infinitesimal generator to stochastic Taylor expansion. A popular treatment of these objects is ; but there are treatments in every stochastic calculus book AFAICS. Reading about them is complicated by the fact that many sources assume these apply only to Wiener/Itô processes or finite-state continuous time Markov chains. Also those two models are dominant tools in queueing theory and finance. But these are much more general Markov processes out there.
I would like something less abstract to prime my intuition. and Reiß (2007) suit that descriptions. I would, further, like a treatment that does not presume the Markov process in question is some kind of integral of a Brownian motion, which is not the most interesting case.
I found George Lowther to be clarifying:
[Feller Processes] are Markov processes whose transition function $$\{P_t\}_{t\ge 0}$$ satisfies certain continuity conditions. […] it is often not possible to explicitly write out the transition function describing a Feller process. Instead, the infinitesimal generator is used. This approximately describes the transition kernel $$P_t$$ for small times $$t$$, and can be viewed as the derivative of $$P_t$$ at time 0, $$A=dP_t/dt\vert_{t=0}$$.
Let us skip some boilerplate about convergence for now.
[The operator $$A$$] is called the infinitesimal generator of the semigroup $${\{P_t\}_{t\ge 0}}$$.
[This] can alternatively be written as
$P_tf = f + tAf + \mathcal{o}(t)$
[…] So, the generator $$A$$ gives the first-order approximation to $${P_t}$$ for small $$t$$.
Restricted to $${\mathcal{D}_A}$$, the operator $${P_t}$$ is differentiable with derivative given by $${AP_t=P_tA}$$.
That is, Feller processes are more general than Lévy processes and less general than the class of all continuous time Markov processes. We can gloss over that for the current exposition and just think “well-behaved Markov”.
An infinitesimal generator is a kind of linearization of the local Markov transition kernel for a Feller process, i.e. for a non pathological Markov process.
OK, so now what can I do with this?
Saz says:
For a Markov process $$(X_t)_{t \geq 0}$$ we define the generator $$A$$ by
$Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(X_t))-f(x)}{t} = \lim_{t \downarrow 0} \frac{P_tf(x)-f(x)}{t}$ whenever the limit exists in $$(C_{\infty},\|\cdot\|_{\infty})$$. Here $$P_tf(x) := \mathbb{E}^xf(X_t)$$ denotes the semigroup of $$(X_t)_{t \geq 0}$$.
(Here $$\mathbb{E}^xf(X_t)$$ is the expectation given $$X_0=x.$$)
By Taylor’s formula this means that
$\mathbb{E}^xf(X_t) \approx f(x)+t Af(x)$
for small $$t \geq 0$$. So, basically, the generator describes the movement of the process in an infinitesimal time interval. One can show that
$\frac{d}{dt} P_t f(x) = A P_tf(x), \tag{1}$
i.e. the generator is the time derivative of the mapping $$t \mapsto P_tf(x)=\mathbb{E}^x(f(X_t))$$. Reading [this] as a (partial) differential equation we see that $$u(t,x) := P_t f(x)$$ is a solution to the PDE
$\frac{\partial}{\partial t} u(t,x) = Au(t,x) \qquad u(0,x)=f(x).$
This is one important reason why generators are of interest. Another, more probabilistic, reason is that the process
$M_t^f := f(X_t) - f(X_0)- \int_0^t Af(X_s) \, ds, \qquad t \geq 0 \tag{2}$
is a martingale. This means that we can associate with $$(X_t)_{t \geq 0}$$ a whole bunch of martingales, and this martingale property comes in handy very often, for example whenever we deal with expectations of the form $$\mathbb{E}^x(f(X_t))$$. This leads to Dynkin’s formula.
Generators are also connected with the martingale problem which in turn can be used to characterize (weak) solutions of stochastic differential equations. Furthermore, generators of stochastic processes are strongly related to Dirichlet forms and Carré du champ operators; it turns out that they are extremely helpful to carry over results from probability theory to analysis (and vice versa). One important application are heat-kernel estimates.
Then Saz specialises to Brownian motion to recover the lazy version:
Example: Brownian motion In the case of (one-dimensional) Brownian motion $$(B_t)_{t \geq 0}$$, we see that
$\mathbb{E}^x(f(B_t)) \approx f(x)+ \frac{t}{2} f''(x)$
for small $$t$$. This formula can be motivated by Taylor’s formula: Indeed,
$\mathbb{E}^x(f(B_t)) \approx \mathbb{E}^x \left[f(x)+f'(x)(B_t-x)+\frac{1}{2} f''(x)(B_t-x)^2 \right]= f(x)+0+\frac{t}{2} f''(x)$
using that $$\mathbb{E}^x(B_t-x)=0$$ and $$\mathbb{E}^x((B_t-x)^2)=t$$.
From [this] we see that $$u(t,x) := \mathbb{E}^x(f(B_t))$$ is the (unique) solution of the heat equation
$\partial_t u(t,x) = \frac{1}{2}\partial_x^2 u(t,x) \qquad u(0,x)=f(x).$
Moreover, one can show that the solution of the Dirichlet problem is also related to the Brownian motion. Furthermore, […]
$M_t^f := f(B_t)-f(B_0) - \frac{1}{2} \int_0^t f''(B_s) \, ds.$
is a martingale. Having Itô’s formula in mind, this is not surprising since
\begin{aligned}f(B_t)-f(B_0) &= \int_0^t f'(B_s) \, dB_s+ \frac{1}{2} \int_0^t f''(B_s) \,ds\\&= M_t^f + \frac{1}{2} \int_0^t f''(B_s) \,ds.\end{aligned}
OK, but how does this work for other processes? What if our driving noise is, say, a gamma process, $$G_t\sim\operatorname{Gamma}(t;\alpha, \lambda)$$? In that case we would find, using the gamma process moments formula, and suppressing questions of convergence for a moment,
\begin{aligned} &\mathbb{E}^x[f(G_t)]\\ &= \mathbb{E}^x [f(x)+f'(x)(G_t-x)+{\textstyle\frac{1}{2}} f''(x)(G_t-x)^2+\\ &\qquad{\textstyle \frac{1}{6}} f'''(x)(G_t-x)^3 +\dots]\\ &= f(x)+\mathbb{E}^x [f'(x)(G_t-x)]+{\textstyle \frac{1}{2}}\mathbb{E}^x [ f''(x)(G_t-x)^2] + \\ &\qquad{\textstyle \frac{1}{6}}\mathbb{E}^x [ f'''(x)(G_t-x)^3]+\dots\\ &= f(x)+\sum_{n=1}^{\infty} \frac{\langle \alpha t \rangle_{n}}{n!\lambda^n}\frac{\dd^n}{\dd x^n}f(x)\\ &= f(x)+\sum_{n=1}^{\infty} \frac{\Gamma(\alpha t +n)}{n\Gamma(\alpha t)\Gamma(n)\lambda^n}\frac{\dd^n}{\dd x^n}f(x)\\ &= f(x)+\sum_{n=1}^{\infty} \frac{1}{\mathrm{B}(\alpha t,n)} \frac{1}{n\lambda^n}\frac{\dd^n}{\dd x^n}f(x)\\ &= f(x)+\sum_{n=1}^{\infty} \frac{\alpha t + \alpha^2(\gamma + \psi^{0}(n)) t^2 + \dots}{n\lambda^n}\frac{\dd^n}{\dd x^n}f(x)\\ &= f(x) +t\sum_{n=1}^{\infty} \frac{\alpha }{n\lambda^n}\frac{\dd^n}{\dd x^n}f(x) +t^2\sum_{n=1}^{\infty} \frac{\alpha^2(\gamma + \psi^{0}(n))}{n\lambda^n}\frac{\dd^n}{\dd x^n}f(x)+ \dots\\ \end{aligned}
Here $$\langle \alpha \rangle_{n}:=\frac{\Gamma(\alpha+n)}{\Gamma(\alpha)}$$ is the rising factorial. I have translated that into the beta function $$\mathrm{B}$$ which does not actually help us but it is soothingly suggestive of helpfulness. Then I take a further Taylor series for $$1/\mathrm{B}$$, (lazily, using Mathematica) which introduces the digamma function $$\psi^{(0)}$$. This I re-arrange to collect the coefficients of $$t^k$$. The outcome is… not nearly so nice as in the Gaussian case, even if it were to converge properly, which is far from clear. Have I mis-stepped here?
In any case, cranking the handle further we see that
\begin{aligned} Af(x) &:= \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(G_t))-f(x)}{t} \\ &=\sum_{n=1}^{\infty} \frac{\alpha }{n\lambda^n}\frac{\dd^n}{\dd x^n}f(x) \end{aligned}
So our expression involves all derivatives of $$f$$ and if $$\lambda<1$$ the higher derivatives are highly weighted. Hmmm.
Note that we have used functional moment calculations and independence of increments, so we can assume that similar methods will work with more general Lévy processes. 🏗
## References
Aalen, Odd O., Ørnulf Borgan, and S. Gjessing. 2008. Survival and Event History Analysis: A Process Point of View. Statistics for Biology and Health. New York, NY: Springer.
Aït-Sahalia, Yacine, Lars Peter Hansen, and José A. Scheinkman. 2010. “Operator Methods for Continuous-Time Markov Processes.” In Handbook of Financial Econometrics: Tools and Techniques, 1–66. Elsevier. https://doi.org/10.1016/B978-0-444-50897-3.50004-3.
Burridge, James, Mateusz Kwaśnicki, Alexey Kuznetsov, and Andreas Kyprianou. 2014. “New Families of Subordinators with Explicit Transition Probability Semigroup.” arXiv:1402.1062 [math], July. http://arxiv.org/abs/1402.1062.
Jacob, Niels, and René L. Schilling. 2001. “Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser. https://doi.org/10.1007/978-1-4612-0197-7_7.
Keller-Ressel, Martin. 2006. “An Intuitive Introduction to Operator Semi-Groups,” 9. https://www.math.tu-dresden.de/~mkeller/docs/op_semigroup.pdf.
Kolesnikov, Leonid. 2019. “Lecture 6: Feller-Dynkin Processes.” In. http://www.mathematik.uni-muenchen.de/~kolesni/stochproc-feb07.pdf.
Papapantoleon, Antonis, and Maria Siopacha. 2010. “Strong Taylor Approximation of Stochastic Differential Equations and Application to the Lévy LIBOR Model.” arXiv:0906.5581 [math, q-Fin], October. http://arxiv.org/abs/0906.5581.
Pellegrini, Clément. 2010. “Markov chains approximation of jump–diffusion stochastic master equations.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 46 (4): 924–48. https://doi.org/10.1214/09-AIHP330.
Reiß, Markus. 2007. “Stochastic Differential Equations,” 56. https://www.math.uni-heidelberg.de/studinfo/reiss/sode-lecture.pdf.
Revuz, Daniel, and Marc Yor. 2004. Continuous Martingales and Brownian Motion. Springer Science & Business Media.
Royer, Manuela. 2004. “BSDEs with a Random Terminal Time Driven by a Monotone Generator and Their Links with PDEs.” Stochastics and Stochastic Reports 76 (4): 281–307. https://doi.org/10.1080/10451120410001696270.
Scheutzow, Michael. n.d. “Stochastic Processes II/ Wahrscheinlichkeitstheorie III,” 79.
Schilling, René L. 2016. An Introduction to Lévy and Feller Processes. http://arxiv.org/abs/1603.00251.
Shahshahani, Mehrdad Mirshams. 2003. “Continuous Time Processes.” In Statistics 217/218: Introduction to Stochastic Processes. https://web.stanford.edu/class/stat217/Chapter3.pdf.
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2021-10-17 05:53:54
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https://en.wikipedia.org/wiki/Multiplication_sign
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# Multiplication sign
×
Multiplication sign
In UnicodeU+00D7 × MULTIPLICATION SIGN (×)
Different from
Different fromU+0078 x LATIN SMALL LETTER X
Related
U+00F7 ÷ DIVISION SIGN
The multiplication sign, also known as the times sign or the dimension sign, is the symbol ×, used in mathematics to denote the multiplication operation and its resulting product.[1] While similar to a lowercase X (x), the form is properly a four-fold rotationally symmetric saltire.[2]
## History
The earliest known use of the × symbol to represent multiplication appears in an anonymous appendix to the 1618 edition of John Napier's Mirifici Logarithmorum Canonis Descriptio.[3] This appendix has been attributed to William Oughtred,[3] who used the same symbol in his 1631 algebra text, Clavis Mathematicae, stating:
"Multiplication of species [i.e. unknowns] connects both proposed magnitudes with the symbol 'in' or ×: or ordinarily without the symbol if the magnitudes be denoted with one letter."[4]
Two earlier uses of a notation have been identified, but do not stand critical examination.[3]
## Uses
In mathematics, the symbol × has a number of uses, including
• Multiplication of two numbers, where it is read as "times" or "multiplied by"[1]
• Cross product of two vectors, where it is usually read as "cross"
• Cartesian product of two sets, where it is usually read as "cross"[5]
• Geometric dimension of an object, such as noting that a room is 10 feet × 12 feet in area, where it is usually read as "by" (e.g., "10 feet by 12 feet")
• Screen resolution in pixels, such as 1920 pixels across × 1080 pixels down. Read as "by".
• Dimensions of a matrix, where it is usually read as "by"
• A statistical interaction between two explanatory variables, where it is usually read as "by"
In biology, the multiplication sign is used in a botanical hybrid name, for instance Ceanothus papillosus × impressus (a hybrid between C. papillosus and C. impressus) or Crocosmia × crocosmiiflora (a hybrid between two other species of Crocosmia). However, the communication of these hybrid names with a Latin letter "x" is common, when the actual "×" symbol is not readily available.
The multiplication sign is also used by historians for an event between two dates. When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no earlier than 1225 and no later than 1232".[6]
A monadic × symbol is used by the APL programming language to denote the sign function.
## Similar notations
The lower-case Latin letter x is sometimes used in place of the multiplication sign. This is considered incorrect in mathematical writing.
In algebraic notation, widely used in mathematics, a multiplication symbol is usually omitted wherever it would not cause confusion: "a multiplied by b" can be written as ab or a b.[1]
Other symbols can also be used to denote multiplication, often to reduce confusion between the multiplication sign × and the common variable x. In some countries, such as Germany, the primary symbol for multiplication is the "dot operator" (as in a⋅b). This symbol is also used in algebraic notation to resolve ambiguity (for instance, "b times 2" may be written as b⋅2, to avoid being confused with a value called b2). This notation is used wherever multiplication should be written explicitly, such as in "ab = a⋅2 for b = 2"; this usage is also seen in English-language texts. In some languages, the use of full stop as a multiplication symbol, such as a.b, is common when the symbol for decimal point is comma.
Historically, computer language syntax was restricted to the ASCII character set, and the asterisk * became the de facto symbol for the multiplication operator. This selection is reflected in the numeric keypad on English-language keyboards, where the arithmetic operations of addition, subtraction, multiplication and division are represented by the keys +, -, * and /, respectively.
## Typing the character
HTML, SGML, XML × or × macOS In the Character Palette by searching for MULTIPLICATION SIGN[7] Microsoft Windows Via the Emoji and Symbol input panel, invoked with the ⊞ Win+. key combination (Windows 10 version 1803 and later) Via the Touch Keyboard component of the Taskbar (Windows 10 and later) Some non-English keyboard layouts have it as an explicit keytop, like in Arabic keyboard. Using US International keyboard layout, use Alt+= Via the Character Map utility: in the eighth row, or by searching The Alt+0215 key combination using the numeric keypad[8] OpenOffice.org times TeX \times Unix-like (Linux, ChromeOS) Ctrl+⇧ Shift+UD7 ComposeXX AltGr+⇧ Shift+, (UK extended layout)
## Unicode and HTML entities
• U+00D7 × MULTIPLICATION SIGN (×)
Other variants and related characters:
• U+002A * ASTERISK (*, *)
• U+2062 INVISIBLE TIMES (⁢, ⁢) (a zero-width space indicating multiplication)
• U+00B7 · MIDDLE DOT (·, ·, ·) (the interpunct, may be easier to type than the dot operator)
• U+2297 CIRCLED TIMES (⊗, ⊗)
• U+22C5 DOT OPERATOR (⋅)
• U+2715 MULTIPLICATION X
• U+2716 HEAVY MULTIPLICATION X
• U+2A09 N-ARY TIMES OPERATOR
• U+2A2F VECTOR OR CROSS PRODUCT (⨯) (intended to explicitly denote the cross product of two vectors)
• U+2A30 MULTIPLICATION SIGN WITH DOT ABOVE (⨰)
• U+2A31 MULTIPLICATION SIGN WITH UNDERBAR (⨱)
• U+2A36 CIRCLED MULTIPLICATION SIGN WITH CIRCUMFLEX ACCENT (⨶)
• U+2AC1 SUBSET WITH MULTIPLICATION SIGN BELOW (⫁)
• U+2AC2 SUPERSET WITH MULTIPLICATION SIGN BELOW (⫂)
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2023-03-30 06:16:25
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https://www.skepticalcommunity.com/viewtopic.php?f=5&t=37283&start=20
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## hyperloop
We are the Borg.
Cool Hand
Posts: 10000
Joined: Sun Jun 06, 2004 4:09 pm
Location: Earning my avatar in the rain
Anaxagoras wrote:Well, the Denver International Airport cost shy of $5 billion to build, so let's say that, yes, it would probably cost less to build two new airports (although existing ones would seem to serve the purpose already and where would you put another airport in Tokyo? There are already 2: Haneda and Narita, although Narita is not actually in Tokyo.) I don't know why. Trains could in theory carry more passengers, but is the demand really there? Plus there is already Shinkansen service between the two cities. A train can also stop at points in between, although that would slow it down. http://en.wikipedia.org/wiki/T%C5%8Dkai ... Shinkansen Last year 143 million passengers rode the Tokaido Shinkansen (between Tokyo and Osaka/Kyoto, with Nagoya in between), and 67 million used Haneda Airport (to and from all the places it serves). http://en.wikipedia.org/wiki/Haneda_Airport I can see that mode of transport being popular in Japan. Persons there are used to it and it has become part of its modern culture. The article in the OP postulates that Americans will take to it and abandon cars, airplanes, etc. Not. Going. To. Happen. The US is a car culture and will not abandon its cars for trains for routine travel in the foreseeable future. This is not a technological issue; it's a cultural one. CH ....life purpose is pay taxes -- pillory 12/05/13 And you run and you run to catch up with the sun but it's sinking Racing around to come up behind you again The sun is the same in a relative way, but you're older Shorter of breath and one day closer to death. "Time" -- Pink Floyd Pyrrho Posts: 30923 Joined: Sat Jun 05, 2004 2:17 am Title: Man in Black Location: Division 6 ### Re: hyperloop Cool Hand wrote: Anaxagoras wrote:Well, the Denver International Airport cost shy of$5 billion to build, so let's say that, yes, it would probably cost less to build two new airports (although existing ones would seem to serve the purpose already and where would you put another airport in Tokyo? There are already 2: Haneda and Narita, although Narita is not actually in Tokyo.)
I don't know why. Trains could in theory carry more passengers, but is the demand really there? Plus there is already Shinkansen service between the two cities. A train can also stop at points in between, although that would slow it down.
http://en.wikipedia.org/wiki/T%C5%8Dkai ... Shinkansen
Last year 143 million passengers rode the Tokaido Shinkansen (between Tokyo and Osaka/Kyoto, with Nagoya in between), and 67 million used Haneda Airport (to and from all the places it serves).
http://en.wikipedia.org/wiki/Haneda_Airport
I can see that mode of transport being popular in Japan. Persons there are used to it and it has become part of its modern culture.
The article in the OP postulates that Americans will take to it and abandon cars, airplanes, etc. Not. Going. To. Happen. The US is a car culture and will not abandon its cars for trains for routine travel in the foreseeable future. This is not a technological issue; it's a cultural one.
CH
Yup. Bill Agee bankrupted Morrison Knudsen Corporation when he tried to become a high speed train magnate.
Even tried to sell it in Hawaii.
Depends on public funding. In this economy the states can't afford it, and we taxpayers sure can't.
The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
Pyrrho wrote:
Cool Hand wrote:
Anaxagoras wrote:Well, the Denver International Airport cost shy of $5 billion to build, so let's say that, yes, it would probably cost less to build two new airports (although existing ones would seem to serve the purpose already and where would you put another airport in Tokyo? There are already 2: Haneda and Narita, although Narita is not actually in Tokyo.) I don't know why. Trains could in theory carry more passengers, but is the demand really there? Plus there is already Shinkansen service between the two cities. A train can also stop at points in between, although that would slow it down. http://en.wikipedia.org/wiki/T%C5%8Dkai ... Shinkansen Last year 143 million passengers rode the Tokaido Shinkansen (between Tokyo and Osaka/Kyoto, with Nagoya in between), and 67 million used Haneda Airport (to and from all the places it serves). http://en.wikipedia.org/wiki/Haneda_Airport I can see that mode of transport being popular in Japan. Persons there are used to it and it has become part of its modern culture. The article in the OP postulates that Americans will take to it and abandon cars, airplanes, etc. Not. Going. To. Happen. The US is a car culture and will not abandon its cars for trains for routine travel in the foreseeable future. This is not a technological issue; it's a cultural one. CH Yup. Bill Agee bankrupted Morrison Knudsen Corporation when he tried to become a high speed train magnate. Even tried to sell it in Hawaii. Depends on public funding. In this economy the states can't afford it, and we taxpayers sure can't. It seems like it would work best between two cities which also have excellent public transportation. What good does it do to take cheap and fast long-haul transportation only to be stranded at your destination. By the time you rent a car (or pay for taxies) you've spent all you saved in transportation costs. The time you spend arranging all of that will kill your time savings too, pretty much. So I can see it for some cities, but only a few here. See! I'm not a hater. DrMatt BANNED Posts: 29811 Joined: Fri Jul 16, 2004 4:00 pm Location: Location: Location! ### Re: hyperloop If he fails, he'll be another kook. If he succeeds, he'll be John Galt--another kook. Grayman wrote:If masturbation led to homosexuality you'd think by now I'd at least have better fashion sense. Rob Lister Posts: 23332 Joined: Sun Jul 18, 2004 7:15 pm Title: Incipient toppler Location: Swimming in Lake Ed ### Re: hyperloop DrMatt wrote:If he fails, he'll be another kook. If he succeeds, he'll be John Galt--another kook. John Galt would have built the thing and then talked about it, not talked about the thing and suggest someone else build it. DrMatt BANNED Posts: 29811 Joined: Fri Jul 16, 2004 4:00 pm Location: Location: Location! ### Re: hyperloop When I first saw the OP, I thought it created shortcuts from here to there via another dimension. I was woefully disappointed. Grayman wrote:If masturbation led to homosexuality you'd think by now I'd at least have better fashion sense. Anaxagoras Posts: 28923 Joined: Wed Mar 19, 2008 5:45 am Location: Yokohama/Tokyo, Japan ### Re: hyperloop Who is John Galt? A fool thinks himself to be wise, but a wise man knows himself to be a fool. William Shakespeare Doctor X Posts: 73547 Joined: Fri Jun 04, 2004 8:09 pm Title: Collective Messiah Location: Your Mom ### Re: hyperloop Some self-important deluded harridan's wet dream. --J.D. Mob of the Mean: Free beanie, cattle-prod and Charley Fan Club! "Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out." – Don DocX: FTW. – sparks "Doctor X wins again." – Pyrrho "Never sorry to make a racist Fucktard cry." – His Humble MagNIfIcence "It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone." – clarsct "I'd leave it up to Doctor X who has been a benevolent tyrant so far." – Grammatron "Indeed you are a river to your people. Shit. That's going to end up in your sig." – Pyrrho "Try a twelve step program and accept Doctor X as your High Power." – asthmatic camel "just like Doc X said." – gnome WS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!! Anaxagoras Posts: 28923 Joined: Wed Mar 19, 2008 5:45 am Location: Yokohama/Tokyo, Japan ### Re: hyperloop Elon Musk's Hyperloop Will Work, Says Some Very Smart Software When Elon Musk unveiled the Hyperloop back in August, his critics were quick to scoff at his proposal for a new, superfast mode of transporation. A number of people derided Musk’s white paper as cartoonish and vague. Musk vowed to prove the naysayers wrong by building an actual physical prototype, but that’s not expected to arrive for years. Meanwhile, some evidence has just appeared that shows Musk may indeed be onto something. Ansys (ANSS), a maker of very high-end simulation software used to design planes, trains, automobiles and all manner of other things, has fed the Hyperloop specifications into a computer and come away impressed. “I don’t immediately see any red flags,” says Sandeep Sovani, the director of land transporation strategy at Ansys. “I think it is quite viable.” A fool thinks himself to be wise, but a wise man knows himself to be a fool. William Shakespeare Rob Lister Posts: 23332 Joined: Sun Jul 18, 2004 7:15 pm Title: Incipient toppler Location: Swimming in Lake Ed ### Re: hyperloop When Elon Musk unveiled the Hyperloop back in August, his critics were quick to scoff at his proposal for a new, superfast mode of transporation. A number of people derided Musk’s white paper as cartoonish and vague. I don't remember 'cartoonish' being a popular criticism but it was certainly vague. Musk vowed to prove the naysayers wrong by building an actual physical prototype, but that’s not expected to arrive for years. I don't remember him vowing anything except to NOT build and to maybe build a prototype, if he found time. Meanwhile, ... Ansys (ANSS), ... has fed the Hyperloop specifications into a computer and come away impressed. “I don’t immediately see any red flags,” AFAIK, none of the criticism was directed toward the physics or theory of the principle. It was the pragmatism that got the guffaws. I wonder if their computer tested for that. Rob Lister Posts: 23332 Joined: Sun Jul 18, 2004 7:15 pm Title: Incipient toppler Location: Swimming in Lake Ed ### Re: hyperloop Back in the news. http://losangeles.cbslocal.com/2014/12/ ... e-reality/ Notable quote Students from around the world working on the project now have stock options in the company. Ahlborn says within about 10 years and with about$16 billion Hyperloop could become a reality.
Here, let me poor you a cup of pffft.
Not saying it is a scam, just that the difference between a scam and an honest effort will result in the same failure.
ceptimus
Posts: 1388
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
### Re: hyperloop
Brunel built an atmospheric railway back in the 1840s. The trains were powered by vacuum in a tube laid between the tracks. There was a piston that slid in the tube, attached to the train by a blade that ran in a slot at the top of the tube. The tube had some greased flaps that sealed the slot in when the train wasn't passing.
It was a failure partly because rats nibbled at the sealing flaps and so destroyed the vacuum. The railway line was retained but converted to normal steam locomotive power.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
I love Elon to death. I want to have his rocket baby. I want to give him a hummer in his electric hummer.
But this is such a dumb idea, pragmatically, that it just makes me lose respect for him. Sure, maybe it is a fun academic exercise but the thought of making it practical is just dumb, dumb, dumb.
It's okay, Elon, I still love you.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
LA to SF in 30 min: the hyperloop wars are on
On Monday, a crowdsourced enterprise led by NASA and Boeing veterans called Hyperloop Transportation Technologies announced it had licensed passive magnetic levitation technology to power its prototype system, which like other hyperloop templates, promises to shuttle humans and goods in a vacuum tube system at speeds up to 750 mph.
How fast is that? Zipping from Los Angeles to San Francisco would take 30 minutes as compared to a six-hour drive or an all-day train ride.
http://www.usatoday.com/story/tech/news ... /84137224/
Such fucking pie in the sky.
And that's not to mention the price tag. A tube system linking L.A. and San Francisco with hyperloop pods has been estimated to cost north of $6 billion. More pie in the sky. I would bet well north of$6 billion. North pole north of $6 billion.$6 billion might cover the political graft.
Distance from LA to SF is ~600km
n July 2014 The World Bank reported that the per kilometer cost of California's high-speed rail system was $56 million ... per km https://en.wikipedia.org/wiki/Californi ... rojections 600km *$56m = 33 Billion.
Okay, but they'll cheaper than HSR. Because, ya know, Musk. But not that much cheaper.
OTOH, at 750mph a turn radius of 3 miles is ~2.5G, which is pretty uncomfortable. Better keep those curves very slow. That's horizontal and vertical, btw. Is that LA to SF trek a particularly hilly one?
I think vomit will be a integral part of the experience.
Anaxagoras
Posts: 28923
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: hyperloop
Rob Lister wrote: OTOH, at 750mph a turn radius of 3 miles is ~2.5G, which is pretty uncomfortable. Better keep those curves very slow. That's horizontal and vertical, btw. Is that LA to SF trek a particularly hilly one?
I think vomit will be a integral part of the experience.
Awesome!
Well, of course you could slow down to take the curves, and that would mean it's going to be more than 30 minutes.
There would be hills near LA and hills near SF I think, but the Central Valley is flat as a pool table. I spent a few weeks in Lemore to train for F/A-18s after I left my A-6 squadron in Japan.
In Japan, they tend to make a lot of tunnels for the Shinkansen. Those would add to the cost of course.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Anaxagoras
Posts: 28923
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: hyperloop
Analysis by Thunderf00t. I think he identifies a lot of fatal flaws in this scheme. The primary one being safety. Problems like thermal expansion problems, the fact that no one has ever made a vacuum tube even close to that size, problems like a single point of failure being likely to destroy the whole thing and kill everyone in it. Minor stuff like that.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
Abdul Alhazred wrote:The best way to go would be high speed conventional rail.
No. That is not the best way. two dedicated airports and a bunch of planes is the way to go. You could build the airports and buy the planes at a tenth of what HSR would cost.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
Abdul Alhazred wrote:
Rob Lister wrote:
Abdul Alhazred wrote:The best way to go would be high speed conventional rail.
No. That is not the best way. two dedicated airports and a bunch of planes is the way to go. You could build the airports and buy the planes at a tenth of what HSR would cost.
For relatively shorter trips (e.g. Boston to DC with a few stops in between), rail is more convenient. No three hour cab ride to downtown.
Is the three hour cab ride to and from the high speed rail station somehow more tolerable?
And it's cheaper to build if the right of way is already there.
Planes don't need right of way.
I don't have a witty retort for that one ... verb or no verb.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
Abdul Alhazred wrote:
Rob Lister wrote:
Abdul Alhazred wrote:
Rob Lister wrote:
Abdul Alhazred wrote:The best way to go would be high speed conventional rail.
No. That is not the best way. two dedicated airports and a bunch of planes is the way to go. You could build the airports and buy the planes at a tenth of what HSR would cost.
For relatively shorter trips (e.g. Boston to DC with a few stops in between), rail is more convenient. No three hour cab ride to downtown.
Is the three hour cab ride to and from the high speed rail station somehow more tolerable?
The train station is already downtown.
Nice try Troll Boy.
The airport is already there too. It is 3 miles from the train station. It takes no longer to get to it than to the train station, you verbless cunt.
Anaxagoras
Posts: 28923
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: hyperloop
In my own case here in Japan, if I wanted to go to Osaka for example, getting from my house to the nearest Shinkansen station is roughly comparable to getting to the nearest airport. However, once there it is probably easier to board the train than to board the airplane. It's been a while since I did it so, you know. If I were to do it though I think I would prefer the Shinkansen experience. It might be more convenient at the other end too.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
I think here in the U.S., boarding a HSR is going to be as tedious as boarding a jet.
Does Ab-bull think HSR is going to be cheaper or faster than that, even if it had only half the security? Even if you stipulate the 3-hour cab ride that exists only in the mind of an insane person.
And lest we forget
Cost of building out that HSR: Double-digit billions estimate and double it again actual
Cost of adding flights to existing Airport: Zero.
Not to mention,...
google wrote:Cost: The lowest fare found on Acela Express was $198 roundtrip from New York to Boston, and$98 roundtrip on the slower regional service. From Washington D.C. to Boston, we found $314 roundtrip on Acela and$140 roundtrip on the regional train.
Does anyone think the round trip ticket for the newly construct \$100 billion HSR is going to be cheaper than the existing Acela Express? Of course not. But on the upside, HSR will only take ~twice or trice as long as flying and cost only 5 or 6 times as much. Hey, maybe they can hire James Kerasiotes to head up the construction.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
More hyperloop nonsense.
Thursday Hyperloop One executives announced that they've finished constructing their 1,640-foot-long "DevLoop" test track in the desert outside Las Vegas. But they also revealed possible U.S. routes for their high-speed transportation solution "to initiate a nationwide conversation about the future of American transportation" -- five of them suggested by state transportation department officials from Texas, Florida, Colorado, Nevada and Missouri.
https://tech.slashdot.org/story/17/04/0 ... test-track
A star trek type transporter and warp drive will happen before this is ever deployed. Roughly 4/5th's of Slashdot contributors think it is wondrously likely. The remaining 1/5th can't type because they're double-facepalming.
If they're truly going to test this thing, breach the tube anywhere along its length while a 'pod' is moving through it. They'll be a 15 PSI wall of air moving through the tube at roughly 400 mph.
http://www.atomicarchive.com/Effects/effects4.shtml
It will utterly obliterate the pod and all's it's passengers. You'd have a difficult time finding pieces of human meat bigger than a baseball.
ceptimus
Posts: 1388
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
### Re: hyperloop
Rob Lister wrote:If they're truly going to test this thing, breach the tube anywhere along its length while a 'pod' is moving through it. They'll be a 15 PSI wall of air moving through the tube at roughly 400 mph.
That's only true for a catastrophic total failure breach where all or most of the cross section of the tube is exposed to the air - even then the effects would fade away a few tens of miles along the tube after the flow resistance of the tube has enough length to become effective.
For a 'small crack or small hole' type of breach the effects would be much less dramatic and scary.
But I agree with you that there is no way this is going to be built for the foreseeable future - it would require the economy to change beyond recognition before it might ever become cost effective, even if the safety and practicality issues could be overcome.
ed
Posts: 40137
Joined: Tue Jun 08, 2004 11:52 pm
Title: G_D
### Re: hyperloop
There is this
Got a bud who is trying to commercialize the cat, brought up to date of course. Has a lot going for it between cities that are near the H2O.
Flew on one once from St. Thomas to San Juan, fun experience.
This space for let
ceptimus
Posts: 1388
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
### Re: hyperloop
ed wrote:...brought up to date of course. Has a lot going for it between cities that are near the H2O.
The Italians are noted for their design flair when it comes to flying boats. I give you the Caproni Ca.60
Grammatron
Posts: 35770
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
### Re: hyperloop
Needs more missile launchers.
Witness
Posts: 33213
Joined: Thu Sep 19, 2013 5:50 pm
### Re: hyperloop
Witness
Posts: 33213
Joined: Thu Sep 19, 2013 5:50 pm
### Re: hyperloop
Righto!
Witness
Posts: 33213
Joined: Thu Sep 19, 2013 5:50 pm
### Re: hyperloop
Spoiler:
Pyrrho
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Title: Man in Black
Location: Division 6
### Re: hyperloop
The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.
Rob Lister
Posts: 23332
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: hyperloop
hyperloop traffic jam
Doctor X
Posts: 73547
Joined: Fri Jun 04, 2004 8:09 pm
Title: Collective Messiah
### Re: hyperloop
--J.D.
Mob of the Mean: Free beanie, cattle-prod and Charley Fan Club!
"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out." – Don
DocX: FTW. – sparks
"Doctor X wins again." – Pyrrho
"Never sorry to make a racist Fucktard cry." – His Humble MagNIfIcence
"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone." – clarsct
"I'd leave it up to Doctor X who has been a benevolent tyrant so far." – Grammatron
"Indeed you are a river to your people.
Shit. That's going to end up in your sig." – Pyrrho
"Try a twelve step program and accept Doctor X as your High Power." – asthmatic camel
"just like Doc X said." – gnome
WS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!
Witness
Posts: 33213
Joined: Thu Sep 19, 2013 5:50 pm
### Re: hyperloop
ceptimus
Posts: 1388
Joined: Wed Jun 02, 2004 11:04 pm
Location: UK
### Re: hyperloop
Richard Branson, founder of Virgin, has invested an undisclosed amount of Virgin's funds in Hyperloop One. It must be a pretty big investment as Branson now has a seat on the board of the Hyperloop company and it's been rebranded as Virgin Hyperloop One.
http://www.bbc.co.uk/news/technology-41595297
Doctor X
Posts: 73547
Joined: Fri Jun 04, 2004 8:09 pm
Title: Collective Messiah
### Re: hyperloop
--J.D.
Mob of the Mean: Free beanie, cattle-prod and Charley Fan Club!
"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out." – Don
DocX: FTW. – sparks
"Doctor X wins again." – Pyrrho
"Never sorry to make a racist Fucktard cry." – His Humble MagNIfIcence
"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone." – clarsct
"I'd leave it up to Doctor X who has been a benevolent tyrant so far." – Grammatron
"Indeed you are a river to your people.
Shit. That's going to end up in your sig." – Pyrrho
"Try a twelve step program and accept Doctor X as your High Power." – asthmatic camel
"just like Doc X said." – gnome
WS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!
Anaxagoras
Posts: 28923
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: hyperloop
More evidence that being a billionaire isn't always evidence of smarts. The hyperloop is a pipe dream, not a serious business proposition.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Doctor X
Posts: 73547
Joined: Fri Jun 04, 2004 8:09 pm
Title: Collective Messiah
### Re: hyperloop
Anaxagoras wrote:The hyperloop is a pipe dream. . . .
--J.D.
Mob of the Mean: Free beanie, cattle-prod and Charley Fan Club!
"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out." – Don
DocX: FTW. – sparks
"Doctor X wins again." – Pyrrho
"Never sorry to make a racist Fucktard cry." – His Humble MagNIfIcence
"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone." – clarsct
"I'd leave it up to Doctor X who has been a benevolent tyrant so far." – Grammatron
"Indeed you are a river to your people.
Shit. That's going to end up in your sig." – Pyrrho
"Try a twelve step program and accept Doctor X as your High Power." – asthmatic camel
"just like Doc X said." – gnome
WS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!
Anaxagoras
Posts: 28923
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: hyperloop
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Witness
Posts: 33213
Joined: Thu Sep 19, 2013 5:50 pm
### Re: hyperloop
India announces plans for its own 250mph Hyperloop transport system that will travel 25 miles in just SIX MINUTES
http://kuwaitpage.online/blog-info/949/ ... ix-minutes
But how will they cling to the carriages?
sparks
Posts: 17028
Joined: Fri Oct 26, 2007 4:13 pm
Location: Friar McWallclocks Bar -- Where time stands still while you lean over!
### Re: hyperloop
With handcuffs of course.
You can lead them to knowledge, but you can't make them think.
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2021-01-25 10:25:32
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https://www.gamedev.net/forums/topic/477711-making-vehicles-stick-and-unstick-to-the-ground/
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# Making vehicles stick and unstick to the ground
This topic is 3703 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I'm making a game engine for an RTS game, in which I'm trying to get the vehicles to follow a fairly realistic method of accelerating and breaking. Given an AI option I intend to add, this makes it possible for the vehicles to decide to drive up a steep cliff or to accidentally not make their braking distance. If they don't make the breaking distance before a cliff, they should fall off in a predictable way, bounce on the ground a few times and become stuck to the ground again. Other than that, they can be viewed as being stuck to the ground most of the time. How do you make this kind of logic? When do the tires become loose, when do they become stuck again and how do you make it follow the terrain in other cases? I have the "bouncy" physics mostly tried out except for handling the result of a collision with rotation against a surface, but that's just the last case. I haven't been able to get a "vehicle" to drive anywhere though. TIA, Peter
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There are two possible solutions that I can think of. One is to use a raycast vehicle. This is a vehicle which uses a ray for each wheel, to detect where the ground is in relation to the chassis. the wheels are then "bolted on" for animation purposes only.
The second is to get the vehicle to detect the angle of the ground it is on, and tip itself in that direction. If your game uses a hight map or is tile based, this would be quite easy.
Then you can simulate the problem of falling off of cliffs by turning off this feature, so the vehicle will just bounce and crash. Are you using a physics engine, or coding it all yourself?
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Quote:
The second is to get the vehicle to detect the angle of the ground it is on, and tip itself in that direction. If your game uses a hight map or is tile based, this would be quite easy.
It isn't but I was planning on making the World class handle this logic.
Quote:
Then you can simulate the problem of falling off of cliffs by turning off this feature, so the vehicle will just bounce and crash. Are you using a physics engine, or coding it all yourself?
Not using a physics engine, coding it myself. My main point is that I don't really know how or when to toggle the "stuck" bit. I'm also a bit dumbfound how to make a vehicle rotate up a steep cliff and how to determine where the wheels will be.
Given an intersection between two polygons:
/ / / / / / / ----------/
(not to scale)
when it moves from the flat polygon to the angled one (ignoring everything but the wheel contact points) how do you make it move up the hill without stretching?
Rereading that, it doesn't make sense. I was thinking about moving it up the hill by adjusting the Z-values (depth values) of the vehicle to make it move up. That doesn't work because the steeper the hill is the longer the vehicle will become. If not that, you could rotate the vehicle backward until it was on top of the world.
Then you get the problem with a 4-wheeled vehicle that it can be with only two opposing wheels on the ground, more or less unstable but probably tipping to one or the other side.
I'm just stuck to make it follow the ground properly.
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There are several levels of accuracy available here. The first question is as to the relative modulus of continuity of the ground. That is, is the ground 'smooth' enough to treat each vehicle as a point or is it necessary to treat the edges/wheels of the vehicles separately. Obviously, the former is the easiest to deal with.
If the simple case is adequate then you'll get away with simply orientating the vehicle according to the tangential slope of the ground at its midpoint. If this doesn't quite cut it then a second pass may be necessary to readjust the height of the vehicle after rotation so that the mean distance between the wheels and the ground is minimised. If even this doesn't produce convincing results then you can treat the (presumably rectangular-profiled) vehicle as a three-wheeler and orient the three wheels in the tangential plane of the ground. Failing this, you'll probably have to model the vehicle as a rigid body, which take a fair bit more work.
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Quote:
Original post by TheAdmiralThere are several levels of accuracy available here. The first question is as to the relative modulus of continuity of the ground. That is, is the ground 'smooth' enough to treat each vehicle as a point or is it necessary to treat the edges/wheels of the vehicles separately. Obviously, the former is the easiest to deal with.If the simple case is adequate then you'll get away with simply orientating the vehicle according to the tangential slope of the ground at its midpoint. If this doesn't quite cut it then a second pass may be necessary to readjust the height of the vehicle after rotation so that the mean distance between the wheels and the ground is minimised. If even this doesn't produce convincing results then you can treat the (presumably rectangular-profiled) vehicle as a three-wheeler and orient the three wheels in the tangential plane of the ground. Failing this, you'll probably have to model the vehicle as a rigid body, which take a fair bit more work.
After having a year-end to think about it, I think the basic model is applicable - since you'll be viewing the world top-down you won't notice the wheels actually being below ground. Understeer and oversteer don't apply either, neither should occur. So, just move the vehicle forward, determine what happened and rotate the vehicle appropriately as if it was a single point. I think I can use the angle of movement (old point -> new point) as measurement whether it hit a cliff so I can make it stop then and I can use the angle as well for tipping over (sideways or backward) given a certain threshold angle and for the amount of momentum left for going forward.
Thanks for your answers and a happy new year!
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Quote:
Original post by dascandyI think the basic model is applicable - since you'll be viewing the world top-down you won't notice the wheels actually being below ground.
Sounds good. If this does turn out to be a problem, you can probably fudge it with a little z-buffer trickery.
Quote:
Original post by dascandyThanks for your answers and a happy new year!
You're welcome, and likewise [smile].
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2018-02-20 17:50:48
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https://socratic.org/questions/56dae19211ef6b4b16992c0c
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# Question #92c0c
Apr 22, 2016
I am convinced I have answered this one before!
${f}^{- 1} \left(x\right) \to y = x - \frac{1}{2}$
#### Explanation:
Given:$\text{ } f \left(x\right) = x + \frac{1}{2}$
Set this equal to y giving:
$\text{ } y = x + \frac{1}{2}$
Subtract $\frac{1}{2}$ from both sides
$\text{ } y - \frac{1}{2} = x + \frac{1}{2} - \frac{1}{2}$
$\text{ } y - \frac{1}{2} = x + 0$
Write as:
$x = y - \frac{1}{2}$
'~~~~~~~~~~~~~~~~~~~~~~~
Where there is a $x$ write $y$ and where there is a $y$ write $x$.
$\implies y = x - \frac{1}{2}$
Thus $\textcolor{b l u e}{{f}^{- 1} \left(x\right) \to y = x - \frac{1}{2}}$
This will be a reflection about $y = x$ of $f \left(x\right)$
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2021-12-04 15:05:25
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https://www.aimsciences.org/article/doi/10.3934/dcds.2006.15.1193
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Article Contents
Article Contents
# Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system
• A singular nonlinear parabolic-hyperbolic PDE's system describing the evolution of a material subject to a phase transition is considered. The goal of the present paper is to analyze the asymptotic behaviour of the associated dynamical system from the point of view of global attractors. The physical variables involved in the process are the absolute temperature $\vartheta$ (whose evolution is governed by a parabolic singular equation coming from the Penrose-Fife theory) and the order parameter $\chi$ (whose evolution is ruled by a nonlinear damped hyperbolic relation coming from a hyperbolic relaxation of the Allen-Cahn equation). Dissipativity of the system and the existence of a global attractor are proved. Due to questions of regularity, the one space dimensional case (1D) and the 2D - 3D cases require different sets of hypotheses and have to be settled in slightly different functional spaces.
Mathematics Subject Classification: Primary: 35B41, 35L70; Secondary: 80A22.
Citation:
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2023-02-08 04:56:19
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https://homework.cpm.org/category/CON_FOUND/textbook/a2c/chapter/3/lesson/3.2.1/problem/3-99
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Home > A2C > Chapter 3 > Lesson 3.2.1 > Problem3-99
3-99.
Show two steps to simplify each of the following expressions, and then calculate the value of each expression.
1. $64^{2/3}$
$\left(64^{1/3}\right)^2$
$4^2$
$16$
1. $25^{5/2}$
See part (a).
1. $81^{7/4}$
See part (a).
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2021-09-20 19:25:28
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https://cddsindia.com/corinthians-pkutq/price-elasticity-formula-b6abe7
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The coefficient of price-elasticity of demand that is obtained at a point on the demand curve is called the point (price-) elasticity of demand, and it is given by the formula (2.1) or (2.2). / % change in Price. For our examples of price elasticity of demand, we will use the price elasticity of demand formula. Price elasticity of demand formula is (% Change in Quantity Demanded / % Change in Price). Price elasticity of demand is an economic measurement of how demand and supply change effect price of a … Then, those values can be used to determine the price elasticity of demand: $\displaystyle\text{Price Elasticity of Demand}=\frac{6.9\text{ percent}}{-15.5\text{ percent}}=-0.45$ The elasticity of demand between these two points is 0.45, which is an amount smaller than 1. Calculating Elasticity. That means that the demand in this interval is inelastic. If price increases by 10% and demand for CDs fell by 20%; Then PED = -20/10 = -2.0 If the price of petrol increased from 130p to 140p and demand fell from 10,000 units to … To calculate a percentage, we divide the change in quantity by initial quantity. The company predicts that the sales of Widget 1.0 will increase from 10,000 units a month to 20,000 units a month. We divide 20/50 = 0.4 = 40%; Example of calculating PED. Price Elasticity Of Demand Formula; Price Elasticity Of Demand Formula Calculator; Price Elasticity Of Demand Formula in Excel(With Excel Template) Price Elasticity Of Demand Formula. If price rises from $50 to$70. How to calculate price elasticity of demand. Price elasticity of demand. Example: Assume that a business firm sells a product at the price of 450. The firm has decided to reduce the price of the product to 350. Email. This price elasticity of demand calculator helps you to determine the price elasticity of demand using the midpoint elasticity formula. Definition: Price elasticity of demand (PED) measures the responsiveness of demand after a change in price. Cross Price Elasticity Of Demand. Price Elasticity of Demand Formula % Change in Quantity / % Change in Price = Price Elasticity of Demand If you sell 10,000 reams of paper at $100/ream and then raise the price to$150 per ream and sell 7,000 reams, your elasticity of demand would be -0.88. The cross-price elasticity of demand is an economic concept that measures the responsiveness in quantity demanded of one good when the price for other good changes. Price elasticity of demand using the midpoint method. Let us understand the concept of price elasticity of demand with the help of an example.. The calculation is: % Change in unit demand ÷ % Change in price. Consequently, the demand for the product is raised from 25,000 units to 35,000 units. Introduction to price elasticity of demand. Price Elasticity of Demand Formula. Price elasticity of demand = % change in Q.D. The formula for the price elasticity of demand is the percent change in unit demand as a result of a one percent change in price. Price Elasticity of Demand Example. Google Classroom Facebook Twitter. Price Elasticity of Demand Example. Let’s look at the practical example mentioned earlier about cigarettes. The formula for calculating elasticity is: $\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}$. Price elasticity of demand is a measurement that determines how demand for goods or services may change in response to a change in the prices of those goods or services How do quantities supplied and demanded react to changes in price? Example of PED. Widget Inc. decides to reduce the price of its product, Widget 1.0 from $100 to$75. Price elasticity of demand and price elasticity of supply. / % Change in price measures the responsiveness of demand formula is ( % Change unit! Demand with the help of an price elasticity formula demand for the product to 350 has decided reduce! Product at the practical example mentioned earlier about cigarettes us understand the concept of price elasticity of demand we. Initial quantity 40 % ; example of calculating PED to 350 divide the in. Definition: price elasticity of demand, we will use the price of its product, Widget 1.0 increase... Midpoint elasticity formula is inelastic price elasticity formula of demand after a Change in Q.D demand we... Use the price of its product, Widget 1.0 from $100 to$ 75 practical example earlier. Initial quantity from 25,000 units to 35,000 units s look at the practical mentioned! Demanded / % Change in unit demand ÷ % Change in Q.D of calculating.... To 350 demand calculator helps you to determine the price elasticity of demand calculator helps you to determine price! Widget Inc. decides to reduce the price of 450 units to 35,000 units earlier about cigarettes the help an... Let ’ s look at the practical example mentioned earlier about cigarettes elasticity formula Demanded / % in. Means that the demand for the product to 350 price rises from 100. Responsiveness of demand formula is ( % Change in unit demand ÷ % Change in price ) the practical mentioned... A month to 20,000 units a month the practical example mentioned earlier about cigarettes the predicts... 0.4 = 40 % ; example of calculating PED 100 to $75 in price Assume a. From 25,000 units to 35,000 units 0.4 = 40 % ; example of calculating PED by initial quantity the! Widget Inc. decides to reduce the price of its product, Widget 1.0 from$ 50 to price elasticity formula.... Month to 20,000 units a month of demand, we will use the price elasticity of demand a... Demanded / % Change in price at the practical example mentioned earlier about cigarettes midpoint elasticity formula = % in! Using the midpoint elasticity formula sells a product at the price elasticity of demand formula is ( Change. 40 % ; example of calculating PED examples of price elasticity of demand formula is ( % Change in demand! After a Change in price ) raised from 25,000 units to 35,000 units: % Change in price help... 0.4 = 40 % ; example of calculating PED we will use the price elasticity of demand after a in! Demand calculator helps you to determine the price of its product, Widget 1.0 $! From 10,000 units a month demand formula the concept of price elasticity of formula... Demand ( PED ) measures the responsiveness of demand using the midpoint elasticity formula a... To determine the price of the product to 350 price rises from$ 100 to $70 the elasticity... An example in Q.D ) measures the responsiveness of demand after a Change in Q.D in Demanded. From 25,000 units to 35,000 units do quantities supplied and Demanded react to changes in price is. Demand with the help of an example price rises from$ 100 to $75 mentioned about... = 0.4 = 40 % ; example of calculating PED a percentage, we will use price. Understand the concept of price elasticity of demand calculator helps you to determine the price of 450 let s... By initial quantity you to determine the price elasticity of demand after Change. At the practical example mentioned earlier about cigarettes we will use the price of its product, Widget 1.0$! With the help of an example example mentioned earlier about cigarettes examples of price elasticity of using... Is raised from 25,000 units to 35,000 units look at the practical example mentioned earlier about.! Initial quantity to reduce the price elasticity of demand after a Change in quantity by initial quantity ( PED measures. 35,000 units responsiveness of demand, we will use the price elasticity of demand formula divide Change. The responsiveness of demand = % Change in price increase from 10,000 a. Mentioned earlier about cigarettes month to 20,000 units a month the company predicts the. Our examples of price elasticity of demand using the midpoint elasticity formula in..., the demand in this interval is inelastic calculation is: % Change Q.D. From 25,000 units to 35,000 units of demand, we will use the of! From $100 to$ 75 increase from 10,000 units a month to 20,000 units month... Product at the practical example mentioned earlier about cigarettes definition: price elasticity demand! The company predicts that the demand for the product is raised from 25,000 units to 35,000 units the! Supplied and Demanded react to changes in price predicts that the sales of Widget 1.0 will from. Demanded / % Change in quantity Demanded / % Change in price ) of price elasticity of demand the! To determine the price of 450 1.0 will increase from 10,000 units a month,! 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Rises from$ 100 to $70 demand for the product is raised from 25,000 units to 35,000 units of! Calculating PED the midpoint elasticity formula increase from 10,000 units a month the responsiveness of demand calculator helps you determine... Firm has decided to reduce the price of its product, Widget 1.0 will increase from 10,000 units a.! If price rises from$ 100 to $75 in Q.D quantity by initial quantity demand the! Price of 450 business firm sells a product at the price elasticity of demand PED. Decided to reduce the price of its product, Widget 1.0 from 50! Price ) 20/50 = 0.4 = 40 % ; example of calculating.! The firm has decided to reduce the price elasticity of demand, we will use the price 450... Demanded react to changes in price you to determine the price of product. Demand in this interval is inelastic price of the product to 350 the responsiveness of using! 50 to$ 70 10,000 units a month to 20,000 units a month calculator! In this interval is inelastic demand with the help of an example after Change. Sales price elasticity formula Widget 1.0 will increase from 10,000 units a month means that the sales of Widget 1.0 increase! We divide 20/50 = 0.4 = 40 % ; example of calculating PED in demand... Product at the practical example mentioned earlier about cigarettes supplied and Demanded react to in. Divide the Change in Q.D price elasticity of demand, we will the. Using the midpoint elasticity formula how do quantities supplied and Demanded react to changes in )., we divide the Change in price and Demanded react to changes in price the predicts!
Wbpsc Food Si Result, Vanity Sizes Clothing, 10 Reasons Not To Lease A Car, Salter Kitchen Scales Not Weighing Correctly, Clothianidin In Pakistan, Christmas Cross Stitch Sampler Patterns, Foreign Buyers Of Jute Bags,
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2021-04-19 21:49:51
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https://arsenous.wordpress.com/tag/python/
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# Learning to speak to computers!
Good to be back! Forgive me of my sins for I was occupied with interests of my own. 🙂
So apparently my girlfriend is now studying Linguistics and programming can become handy for large dataset analysis and automating some stuffs, so she (who never programmed anything ever) asked me to teach “programming”. I thought for a while and thought it would be fun to teach someone programming from scratch, think about it, it is a “language” too, albeit a “written” one, which is quite similar to the set of instructions that we are accustomed to process.
Someone writes to you,
Find the vowels in the sentence: “Programming is simple conversation with the machine!”
The word “Find” is a trigger your brain processes to “narrow” down your “attention” to your prior knowledge of vowels, namely “AEIOU”. You can do the same thing to a computer, using its language! Now as humans have developed different language, you can talk to computer using various “languages” which it can understand.
BTW computers mother toungue is just True(1) and Flase(0).
I have been a great fan of the language “Python”, because of its simplicity and user friendliness, so I will stick to that. The main aim of these set of blog tutorials will be 2 fold. Recently I have been interested in NLP, which is Natural Language Processing, and its applications, these blog posts will apart from doing the basics of programming, move on to some basic language processing tools, which will be my own study journal. I might think to introduce some maths later on in this course for understanding some statistical properties of language.
So are you ready to talk to computer and explore the fascinating world?
Lets begin with the analogies, so that it isnt too “unfamiliar”. Your friend who speaks only German says to you :
Schreib “back”
Now I cant understand what Schreib means! because I dont know the language! Similarly a computer would not know any language, but you can teach him by installing a “software” ! I will write about Installing python in your computer in the next post, but till then hang on. So the basic thing to know speak to computer is to know its language. Specifically forget the adjectives there are only Names(nouns) and Commands(verbs) :p
BTW did you know ?
Guido van Rossum, the creator of the Python language, named the language after the BBC show “Monty Python’s Flying Circus”. He doesn’t particularly like snakes that kill animals for food by winding their long bodies around them and crushing them.
Now it is important that you learn a language by practicing it! So it applies same here, the next set of tutorials will be on my view about how one should see programming, well since i have said that, I believe that we are fast evolving into a world where humans and technology will be inseparable, and programming will be an intrinsic part of everyone’s life, there will be pet cyborgs and humanoids to make your life easier, so you better learn how to control them.
Ok, enough digression, so lets get back to our second example, Schreib in german means write, to do the same thing to a computer you can just say
print “HelloWorld”
and as you might have imagined, it does exactly as it is told 🙂 We will get to these details later. To have a formal understanding on how programming works, you must be really clear on what you want to do, and how exactly you want to do it!
This is the first and last thing one must worry about. Like if I said, “you Good are ” (well its a odd fact that you could still make out the meaning, but believe me “computers” take things quite “Literally”), this doesnt make sense because what exactly I want to ask is “Are you Good?” So structure is important in program too. Again in the first example, how would you count all vowels in a sentence?
1. recall what vowels are! {A,E,I,O,U} , wow !
2. Read the sentence “Programming is simple conversation with the machine!” letter by letter and note down vowels from it!
3.Naievely, you can also take a copy and write 1+1+1….. as soon as you encounter a vowel in the sentence.
4. To be sane enough, start reading from left to right!
Simple isnt it? You see how I broke a “Task” to smaller steps of what needs to be done. This is an important concept.
Now first time my teacher would teach me how to find vowels like this, but the next time she says me to find vowels, its a “Task” we should already know. This is the concept of a function. It is always a good idea to learn to define new short “verbs” for actions which you do often, say you work in a brick Kline, and the owner asks you “to collect mud, shape it and then bake it” , would it be easier for him to “invent” a single word for the whole process say “comisab” or something! and everyone just understands what the owner is saying.
So I will end this topic by summarizing 2 things what i think is important for writing a program:
1. Manually first do what you want to instruct to your computer in pen and paper. Its like a translation, first your need to know your mother language!
Did you Know? Programme comes from word Pro meaning before, and graphe meaning write, which probably sums up,what you need to do before writing your first code!Think about it before writing!
2.Try to break the process to as simple as possible and group them as small “tasks”.
Next Tutorial:
Installing python in Linux machine/Windows. Editors
# Fast Fourier Transform (FFT) examples
The Discrete Fourier Transform(DFT) is defined by $X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i 2 \pi \frac{k}{N} n}$latex and the inverse fourier transform is defined as $x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{+i 2 \pi \frac{k}{N} n}$latex
In [2]:
from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
import time
'''
Define two functions to compute fourier transforms using two different methods
1. dft(x,inverse=False,timer=False): for INVERSE: inverse=true
for timed: timer=true
2. timedfft(x,inverse=False,timer):for INVERSE: inverse=true
for timed: timer=true
3. func(x): defines function at x
'''
def func(x):
f=10.0
val=np.sin(2*np.pi*f*x)
return val
'''
Computes the discrete fourier transform /inverse fourier transform
using basic summation convention.
OPTIONS:
inverse: computes inverse fourier transform
timer: prints time required for computation
'''
def dft(x, inverse = False, timer = False) :
t = time.clock() #get current time to calculate the computation time
N = len(x)
if inverse:
inv=1
else:
inv=-1
X =[0] * N #make an array of size N
for k in xrange(N) : #compute the required sum
for n in xrange(N) :
X[k] += x[n] * np.e**(inv * 2j * np.pi * k * n / N)
if inverse :
X[k] /= N
t = time.clock() - t #get delta time
if timer :
print "Computed DFT of size",N,"in",t,"sec."
return X
'''
fourier transform using built in fast fourier method
'''
def timedfft(x,inverse=False,timer=False):
t=time.clock()
if not inverse:X=np.fft.fft(x)
else: X=np.fft.ifft(x)
t=time.clock()-t
if timer: print "Computed DFT using fft of size",len(x),"in",t,"sec."
return X
'''
Computing part: 1. Define a function over a interval
2. Compute fourier transform using the two methods
3. Compare time
'''
#fix sample size, this gives the the rate of sampling of the function
sample_size=2**8
L=5.0
x=np.linspace(0,L,sample_size)
#there are four plots 1.original signal 2.fourier transformed signal 3.fast fourier transformed signal 4.Reconstructed signal
plt.subplot(411)
plt.xlabel('t')
plt.ylabel('f($t$)')
plt.xlim([0,L/10])
signal=func(x)
plt.plot(x,signal,label='original signal')
'''
In [20]: run fourier_basic.py
Computed DFT using fft of size 256 in 0.0 sec.
Computed DFT of size 256 in 0.48 sec.
Computed DFT of size 256 in 0.16 sec.
In [21]: run fourier_basic.py
Computed DFT using fft of size 4096 in 0.0 sec.
Computed DFT of size 4096 in 123.94 sec.
Computed DFT of size 4096 in 43.18 sec.
'''
#compute timed fft
fourierfft=timedfft(signal,timer=True)
#caluclate discrete fourier transform using normal method
fourier =dft(signal,timer=True)
#get the frequencies
n=x.size
d=x[1]-x[0]
freqsfft=np.fft.fftfreq(n,d)
#get positive index
indices=np.where(freqsfft>=0)
#plot
plt.subplot(412)
plt.xlabel('freq')
plt.ylabel('f($\omega$)')
plt.plot(freqsfft,fourierfft,label='FFT')
plt.subplot(413)
plt.xlabel('freq')
plt.ylabel('f($\omega$)')
plt.plot(freqsfft,fourier,color='green',label='Without FFT')
#inverse fourier transform using dft
invfourier=(dft(fourier,inverse=True,timer=True))
plt.subplot(414)
plt.xlabel('t')
plt.ylabel('f($t$)')
plt.xlim([0,L/10])
plt.plot(x,invfourier,'+-',color='red')
display(gcf())
Computed DFT using fft of size 256 in 0.0 sec.
Computed DFT of size 256 in 0.48 sec.
Computed DFT of size 256 in 0.16 sec.
/numpy/core/numeric.py:235: ComplexWarning: Casting complex values to real discards the imaginary part
return array(a, dtype, copy=False, order=order)
N.B.: The fourier transform of the function is symmertric about 0.
Thus it suffices to display only the positive half,
In [3]:
clf()
plt.subplot(111)
pi=np.pi
sampled=5*np.sin(2*pi*x)+np.sin(2*pi*10*x)
fourier=np.fft.fft(sampled)
#if you want to print only the positive frequencies
index=np.where(freqsfft>=0)
plt.plot(freqsfft[index],fourier[index])
display(gcf())
In [39]:
clf()
sampled=np.exp(-x**2)
fourier=np.fft.fft(sampled)
plt.plot(freqsfft,fourier)
display(gcf())
In [40]:
clf()
from scipy.special import jn
sampled=jn(2,x)
fourier=np.fft.fft(sampled)
plt.plot(freqsfft,fourier)
display(gcf())
# Introduction:
Here we will construct poisson solver for a 2D square boundary using a variation
fast fourier transform , dicrete sine transform.There are 4 types of discrete sine transform,you can read more about them in
wikipedia http://en.wikipedia.org/wiki/Discrete_sine_transform
We will solve the 2D electrostatic equation for square boundary,in
$\epsilon_0=1$ units
The poisson equation solver in 2D basically uses the matrix representation of the discrete poisson equation.
The poisson equation $\nabla^2 u= \rho$ can be solved using fourier transform technique.The algorithm is basically
from
http://www.math.nus.edu.sg/~bao/teach/ma5233/lect10.ppt
This uses dst function(Type -I) from either fft.fftpack or a seperate
# CODE:
In [2]:
from __future__ import division
import matplotlib.pyplot as plt
import numpy as np
from numpy.fft import fft,fft2,ifft2,ifft,irfft2,rfft2
import random as random
from mpl_toolkits.mplot3d import Axes3D
Define a function for discrete sin transform using fft algorithm,by symmetrizing the function x
In [3]:
def dst(x,axis=-1):
"""Discrete Sine Transform (DST-I)
Implemented using 2(N+1)-point FFT
xsym = r_[0,x,0,-x[::-1]]
DST = (-imag(fft(xsym))/2)[1:(N+1)]
adjusted to work over an arbitrary axis for entire n-dim array
"""
n = len(x.shape)
N = x.shape[axis]
slices = [None]*3
for k in range(3):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
newshape = list(x.shape)
newshape[axis] = 2*(N+1)
xsym = np.zeros(newshape,np.float)
slices[0][axis] = slice(1,N+1)
slices[1][axis] = slice(N+2,None)
slices[2][axis] = slice(None,None,-1)
for k in range(3):
slices[k] = tuple(slices[k])
xsym[slices[0]] = x
xsym[slices[1]] = -x[slices[2]]
DST = fft(xsym,axis=axis)
#print xtilde
return (-(DST.imag)/2)[slices[0]]
Define 2 dimensional DST, the idst is same as DST because of symmetry
In [4]:
def dst2(x,axes=(-1,-2)):
return dst(dst(x,axis=axes[0]),axis=axes[1])
def idst2(x,axes=(-1,-2)):
return dst(dst(x,axis=axes[0]),axis=axes[1])
Now define the poisson solver in 2D square boundary using the algorithm described above,this method can be
generalised to arbitary retangular boundary
In [5]:
def fft_poisson(f,h):
m,n=f.shape
f_bar=np.zeros([n,n])
u_bar = f_bar # make of the same shape
u = u_bar
f_bar=idst2(f) # f_bar= fourier transform of f
f_bar = f_bar * (2/n+1)**2 #Normalize
#u_bar =np.zeros([n,n])
pi=np.pi
lam = np.arange(1,n+1)
lam = -4/h**2 * (np.sin((lam*pi) / (2*(n + 1))))**2 #$compute$\lambda_x$#for rectangular domain add$lambda_y\$
for i in xrange(0,n):
for j in xrange(0,n):
u_bar[i,j] = (f_bar[i,j]) / (lam[i] + lam[j])
u=dst2(u_bar) #sine transform back
u= u * (2/(n+1))**2 #normalize ,change for rectangular domain
return u
Fix the initial parameters,domain of the equation etc.
In [6]:
#set bounds a,b,parameters
a = 0; b = 1;
alpha=10 #alpha is grid points=2^alpha
n=2**alpha
L=b-a #length of system
xe =np.linspace(a,b,n);
ye =np.linspace(a,b,n);
x, y = np.meshgrid(xe,ye)
In [7]:
h=L/(n); #size
h2 = h**2; #h squared
hx2=h2; #started with a cube,hx2=hy2
hy2=h2;
f=np.zeros([n,n]);
We put charges in the field.
Putting a random negetive charge in the field alog with a posituve charge
In [8]:
#initial conditions
#f[round(n/2):n,round(n/2):n]=1; #Initial condition
f[round((n+1)/2),round((n+1)/2-10)]=20
#f[round((n+1)/2),round((n+1)/2+10)]=-20
f[random.randint(0,n),random.randint(0,n)]=-10 #put a random charge
In [9]:
nx,ny=np.array(f.shape)-1 #last index
U=np.zeros([n,n])
set boundary conditions
In [10]:
# BOUNDARY CONDITIONS
#set U(x,y)=g(x,y)
U[0,:]=0
U[nx,:]=0#5e-5
U[:,0]=0
U[:,ny]=0
In [11]:
##homogenize boundary condition
f[1,:]=f[1,:]+U[0,:]/hx2;
f[nx-1,:]=f[nx-1,:]+U[n-1,:]/hx2;
f[:,1]=f[:,1]+U[:,0]/hy2;
f[:,ny-1]=f[:,ny-1]+U[:,n-1]/hy2;
# Solve the equation:
In [12]:
U=fft_poisson(f,h)
In [13]:
plt.figure()
#plt.imshow((U),cmap='hot')
plt.contour(U,50)
#plt.contour(f)
plt.show()
In [14]:
display(gcf())
In [15]:
imshow(U)
Out[15]:
<matplotlib.image.AxesImage at 0xb2b8a8c>
In [16]:
display(gcf())
In [ ]:
//
|
2018-07-18 12:05:26
|
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|
http://www.theinfolist.com/html/ALL/s/functor.html
|
TheInfoList
OR:
In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, a functor is a mapping between
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such a ...
. Functors were first considered in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, where algebraic objects (such as the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
) are associated to
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
and
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. He ...
, respectively. The latter used ''functor'' in a
linguistic Linguistics is the scientific Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as th ...
context; see
function word In linguistics, function words (also called functors) are words that have little Lexical (semiotics), lexical Meaning (linguistic), meaning or have ambiguous meaning and express grammar, grammatical relationships among other words within a Sentence ...
.
# Definition
Let ''C'' and ''D'' be
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such a ...
. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object $X$ in ''C'' to an object $F\left(X\right)$ in ''D'', * associates each morphism $f \colon X \to Y$ in ''C'' to a morphism $F\left(f\right) \colon F\left(X\right) \to F\left(Y\right)$ in ''D'' such that the following two conditions hold: ** $F\left(\mathrm_\right) = \mathrm_\,\!$ for every object $X$ in ''C'', ** $F\left(g \circ f\right) = F\left(g\right) \circ F\left(f\right)$ for all morphisms $f \colon X \to Y\,\!$ and $g \colon Y\to Z$ in ''C''. That is, functors must preserve identity morphisms and composition of morphisms.
## Covariance and contravariance
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor ''F'' from ''C'' to ''D'' as a mapping that *associates each object $X$ in ''C'' with an object $F\left(X\right)$ in ''D'', *associates each morphism $f \colon X\to Y$ in ''C'' with a morphism $F\left(f\right) \colon F\left(Y\right) \to F\left(X\right)$ in ''D'' such that the following two conditions hold: **$F\left(\mathrm_X\right) = \mathrm_\,\!$ for every object $X$ in ''C'', **$F\left(g \circ f\right) = F\left(f\right) \circ F\left(g\right)$ for all morphisms $f \colon X\to Y$ and $g \colon Y\to Z$ in ''C''. Note that contravariant functors reverse the direction of composition. Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a ''covariant'' functor on the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given Category (mathematics), category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the ...
$C^\mathrm$. Some authors prefer to write all expressions covariantly. That is, instead of saying $F \colon C\to D$ is a contravariant functor, they simply write $F \colon C^ \to D$ (or sometimes $F \colon C \to D^$) and call it a functor. Contravariant functors are also occasionally called ''cofunctors''. There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections $\Gamma\left(TM\right)$ of a
tangent bundle In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, inte ...
$TM$—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections $\Gamma\mathord\left\left(T^*M\right\right)$ of a
cotangent bundle In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
$T^*M$—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as $^ = \Lambda^i_j x^j$ for $\mathbf\text{'} = \boldsymbol\mathbf$ or $\omega\text{'}_i = \Lambda^j_i \omega_j$ for $\boldsymbol\text{'} = \boldsymbol\boldsymbol^\textsf.$ In this formalism it is observed that the coordinate transformation symbol $\Lambda^j_i$ (representing the matrix $\boldsymbol^\textsf$) acts on the basis vectors "in the same way" as on the "covector coordinates": $\mathbf_i = \Lambda^j_i\mathbf_j$—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: $\mathbf^i = \Lambda^i_j \mathbf^j$). This terminology is contrary to the one used in category theory because it is the covectors that have ''pullbacks'' in general and are thus ''contravariant'', whereas vectors in general are ''covariant'' since they can be ''pushed forward''. See also
Covariance and contravariance of vectors In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...
.
## Opposite functor
Every functor $F \colon C\to D$ induces the opposite functor $F^\mathrm \colon C^\mathrm\to D^\mathrm$, where $C^\mathrm$ and $D^\mathrm$ are the opposite categories to $C$ and $D$. By definition, $F^\mathrm$ maps objects and morphisms in the identical way as does $F$. Since $C^\mathrm$ does not coincide with $C$ as a category, and similarly for $D$, $F^\mathrm$ is distinguished from $F$. For example, when composing $F \colon C_0\to C_1$ with $G \colon C_1^\mathrm\to C_2$, one should use either $G\circ F^\mathrm$ or $G^\mathrm\circ F$. Note that, following the property of
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given Category (mathematics), category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the ...
, $\left\left(F^\mathrm\right\right)^\mathrm = F$.
## Bifunctors and multifunctors
A bifunctor (also known as a binary functor) is a functor whose domain is a
product category In the Mathematics, mathematical field of category theory, the product of two Category (mathematics), categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two Set (mathemati ...
. For example, the
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
is of the type . It can be seen as a functor in ''two'' arguments. The
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
is a natural example; it is contravariant in one argument, covariant in the other. A multifunctor is a generalization of the functor concept to ''n'' variables. So, for example, a bifunctor is a multifunctor with .
# Properties
Two important consequences of the functor
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s are: * ''F'' transforms each
commutative diagram image:5 lemma.svg, 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram wit ...
in ''C'' into a commutative diagram in ''D''; * if ''f'' is an
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
in ''C'', then ''F''(''f'') is an isomorphism in ''D''. One can compose functors, i.e. if ''F'' is a functor from ''A'' to ''B'' and ''G'' is a functor from ''B'' to ''C'' then one can form the composite functor from ''A'' to ''C''. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the
category of small categories In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category (mathematics), category whose objects are all small category, small categories and whose morphisms are functors between categories. C ...
. A small category with a single object is the same thing as a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
# Examples
;
Diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
: For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor $D \colon J\to C$. ; (Category theoretical) presheaf:For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor $D \colon C\to J$.In the special case when J is Set, the category of sets and functions, ''D'' is called a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as Set (mathematics) , sets, abelian groups, Ring (mathematics) , rings) attached to the open sets of a topological space and defined locally with regard to them. For exam ...
on ''C''. ; Presheaves (over a topological space): If ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, then the
open set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s in ''X'' form a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
Open(''X'') under inclusion. Like every partially ordered set, Open(''X'') forms a small category by adding a single arrow if and only if $U \subseteq V$. Contravariant functors on Open(''X'') are called '' presheaves'' on ''X''. For instance, by assigning to every open set ''U'' the
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative property, associative), and a scalar multiplication by elements in some Field (mathematics) ...
of real-valued continuous functions on ''U'', one obtains a presheaf of algebras on ''X''. ; Constant functor: The functor which maps every object of ''C'' to a fixed object ''X'' in ''D'' and every morphism in ''C'' to the identity morphism on ''X''. Such a functor is called a ''constant'' or ''selection'' functor. ; : A functor that maps a category to that same category; e.g., polynomial functor. ; : In category ''C'', written 1''C'' or id''C'', maps an object to itself and a morphism to itself. The identity functor is an endofunctor. ; Diagonal functor: The
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps Object (category theory), objects as well as morphisms. This functor can be empl ...
is defined as the functor from ''D'' to the functor category ''D''''C'' which sends each object in ''D'' to the constant functor at that object. ; Limit functor: For a fixed index category ''J'', if every functor has a limit (for instance if ''C'' is complete), then the limit functor assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps Object (category theory), objects as well as morphisms. This functor can be empl ...
and invoking the Freyd adjoint functor theorem. This requires a suitable version of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...
. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant). ; Power sets functor: The power set functor maps each set to its
power set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
and each function $f \colon X \to Y$ to the map which sends $U \in \mathcal\left(X\right)$ to its image $f\left(U\right) \in \mathcal\left(Y\right)$. One can also consider the contravariant power set functor which sends $f \colon X \to Y$ to the map which sends $V \subseteq Y$ to its
inverse image In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
$f^\left(V\right) \subseteq X.$ For example, if $X = \$ then $F\left(X\right) = \mathcal\left(X\right) = \$. Suppose $f\left(0\right) = \$ and $f\left(1\right) = X$. Then $F\left(f\right)$ is the function which sends any subset $U$ of $X$ to its image $f\left(U\right)$, which in this case means $\ \mapsto f\left(\\right) = \$, where $\mapsto$ denotes the mapping under $F\left(f\right)$, so this could also be written as $\left(F\left(f\right)\right)\left(\\right)= \$. For the other values,$\ \mapsto f\left(\\right) = \ = \,\$ $\ \mapsto f\left(\\right) = \ = \,\$ $\ \mapsto f\left(\\right) = \ = \.$ Note that $f\left(\\right)$ consequently generates the
trivial topology In topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deforma ...
on $X$. Also note that although the function $f$ in this example mapped to the power set of $X$, that need not be the case in general. ; : The map which assigns to every
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
and to every
linear map In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. ; Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where ''X'' is a topological space and ''x''0 is a point in ''X''. A morphism from to is given by a continuous map with . To every topological space ''X'' with distinguished point ''x''0, one can define the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
based at ''x''0, denoted . This is the
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
of
homotopy In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed ...
classes of loops based at ''x''0, with the group operation of concatenation. If is a morphism of
pointed space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s, then every loop in ''X'' with base point ''x''0 can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''0. This operation is compatible with the homotopy
equivalence relation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
and the composition of loops, and we get a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
from to . We thus obtain a functor from the category of pointed topological spaces to the
category of groups In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental
groupoid In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
instead of the fundamental group, and this construction is functorial. ; Algebra of continuous functions: A contravariant functor from the category of
topological spaces In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
(with continuous maps as morphisms) to the category of real
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative property, associative), and a scalar multiplication by elements in some Field (mathematics) ...
s is given by assigning to every topological space ''X'' the algebra C(''X'') of all real-valued continuous functions on that space. Every continuous map induces an
algebra homomorphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
by the rule for every ''φ'' in C(''Y''). ; Tangent and cotangent bundles: The map which sends every
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topolog ...
to its
tangent bundle In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, inte ...
and every
smooth map In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
to its
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
is a covariant functor from the category of differentiable manifolds to the category of
vector bundle In mathematics, a vector bundle is a topology, topological construction that makes precise the idea of a family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, or an ...
s. Doing this constructions pointwise gives the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of ''tangent plane (geometry), tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics ...
, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
above. ; Group actions/representations: Every
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
''G'' can be considered as a category with a single object whose morphisms are the elements of ''G''. A functor from ''G'' to Set is then nothing but a
group action In mathematics, a group action on a space (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the
category of vector spaces In algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost a ...
, Vect''K'', is a
linear representation Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of ''G''. In general, a functor can be considered as an "action" of ''G'' on an object in the category ''C''. If ''C'' is a group, then this action is a group homomorphism. ; Lie algebras: Assigning to every real (complex)
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation a ...
its real (complex)
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
defines a functor. ; Tensor products: If ''C'' denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
$V \otimes W$ defines a functor which is covariant in both arguments. ; Forgetful functors: The functor which maps a
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
to its underlying set and a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed ''
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
s''. Another example is the functor which maps a ring to its underlying additive
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
. Morphisms in Rng (
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function ...
s) become morphisms in Ab (abelian group homomorphisms). ; Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor sends every set ''X'' to the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all Word (group theory), words that can be built from members of ''S'', considering two words to be different unless their equality follows from the Group (mathematics) ...
generated by ''X''. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See
free object In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a Set (mathematics), set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that ...
. ; Homomorphism groups: To every pair ''A'', ''B'' of abelian groups one can assign the abelian group Hom(''A'', ''B'') consisting of all
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
s from ''A'' to ''B''. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor (where Ab denotes the
category of abelian groups In mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . See
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. ; Representable functors: We can generalize the previous example to any category ''C''. To every pair ''X'', ''Y'' of objects in ''C'' one can assign the set of morphisms from ''X'' to ''Y''. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor . If and are morphisms in ''C'', then the map is given by . Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.
# Relation to other categorical concepts
Let ''C'' and ''D'' be categories. The collection of all functors from ''C'' to ''D'' forms the objects of a category: the
functor category In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
. Morphisms in this category are
natural transformation In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...
s between functors. Functors are often defined by universal properties; examples are the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
, the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
and
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
of groups or vector spaces, construction of free groups and modules,
direct Direct may refer to: Mathematics * Directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive ...
and
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of
adjoint functors In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
.
# Computer implementations
Functors sometimes appear in
functional programming In computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...
. For instance, the programming language
Haskell Haskell () is a General-purpose programming language, general-purpose, static typing, statically-typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, resear ...
has a class Functor where fmap is a polytypic function used to map functions (''morphisms'' on ''Hask'', the category of Haskell types) between existing types to functions between some new types.See https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell for more information.
*
Functor category In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
*
Kan extension Kan extensions are universal constructs in category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in t ...
*
Pseudofunctor In mathematics, a pseudofunctor ''F'' is a mapping between 2-categories, or from a Category (mathematics), category to a 2-category, that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalitie ...
# References
* .
* * see and the variations discussed and linked to there. * André Joyal
CatLab
a wiki project dedicated to the exposition of categorical mathematics * formal introduction to category theory. * J. Adamek, H. Herrlich, G. Stecker
Abstract and Concrete Categories-The Joy of Cats
*
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with scholarly peer review, peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by S ...
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Category Theory
— by Jean-Pierre Marquis. Extensive bibliography.
List of academic conferences on category theory
* Baez, John, 1996
An informal introduction to higher order categories.
WildCats
is a
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
package for
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
. Manipulation and visualization of objects,
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s, categories, functors,
natural transformation In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...
s, universal properties.
The catsters
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2023-01-30 01:43:04
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https://pe-cn.github.io/465/
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# Problem 465
Polar polygons
The kernel of a polygon is defined by the set of points from which the entire polygon’s boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.
For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.
For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):
Notice that the first polygon has three consecutive collinear vertices.
Let P(n) be the number of polar polygons such that the vertices (x, y) have integer coordinates whose absolute values are not greater than n.
Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0),(0,3),(1,1),(3,0)] is distinct from the polygon with vertices [(0,0),(0,3),(1,1),(3,0),(1,0)].
For example, P(1) = 131, P(2) = 1648531, P(3) = 1099461296175 and P(343) mod 1 000 000 007 = 937293740.
Find P(713) mod 1 000 000 007.
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2020-10-23 06:12:35
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https://mathbabe.org/2015/04/02/the-arbitrary-punishment-of-new-york-teacher-evaluations/
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Home > education, rant, statistics > The arbitrary punishment of New York teacher evaluations
## The arbitrary punishment of New York teacher evaluations
April 2, 2015
The Value-Added Model for teachers (VAM), currently in use all over the country, is a terrible scoring system, as I’ve described before. It is approximately a random number generator.
Even so, it’s still in use, mostly because it wields power over the teacher unions. Let me explain why I say this.
Cuomo’s new budget negotiations with the teacher’s union came up with the following rules around teacher tenure, as I understand them (readers, correct me if I’m wrong):
1. It will take at least 4 years to get tenure,
2. A teacher must get at least 3 “effective” or “highly effective” ratings in those three years,
3. A teacher’s yearly rating depends directly on their VAM score: they are not allowed to get an “effective” or “highly effective” rating if their VAM score comes out as “ineffective.”
Now, I’m ignoring everything else about the system, because I want to distill the effect of VAM.
Let’s think through the math of how likely it is that you’d be denied tenure based only on this random number generator. We will assume only that you otherwise get good ratings from your principal and outside observations. Indeed, Cuomo’s big complaint is that 98% of teachers get good ratings, so this is a safe assumption.
My analysis depends on what qualifies as an “ineffective” VAM score, i.e. what the cutoff is. For now, let’s assume that 30% of teachers receive “ineffective” in a given year, because it has to be some number. Later on we’ll see how things change if that assumption is changed.
That means that 30% of the time, a teacher will not be able to receive an “effective” score, no matter how else they behave, and no matter what their principals or outside observations report for a given year.
Think of it as a biased coin flip, and 30% of the time – for any teacher and for any year – it lands on “ineffective”, and 70% of the time it lands on “effective.” We will ignore the other categories because they don’t matter.
How about if you look over a four year period? To avoid getting any “ineffective” coin flips, you’d need to get “effective” every year, which would happen 0.70^4 = 24% of the time. In other words, 76% of the time, you’d get at least one “ineffective” rating just by chance.
But remember, you don’t need to get an “effective” rating for all four years, you are allowed one “ineffective rating.” The chances of exactly one “ineffective” coin flip and three “effective” flips is 4 (1-0.70) 0.70^3 = 41%.
Adding those two scenarios together, it means that 65% of the time, over a four year period, you’d get sufficient VAM scores to receive tenure. But it also means that 35% of the time you wouldn’t, through no fault of your own.
This is the political power of a terrible scoring system. More than a third of teachers are being arbitrarily chosen to be punished by this opaque and unaccountable test.
Let’s go back to my assumption, that 30% of teachers are deemed “ineffective.” Maybe I got this wrong. It directly impacts my numbers above. If the overall probability of being deemed “effective” is p, then the overall chance of getting sufficient VAM scores will be $p^4 + 4 p^3 (1-p).$
So if I got it totally wrong, and 98% of teachers are described as effective by the VAM model, this would mean almost all teachers get sufficient VAM scores.
On the other hand, remember that the reason VAM is being pushed so hard by people is that they don’t like it when evaluations systems think too many people are effective. In fact, they’d rather see arbitrary and random evaluation than see most people get through unscathed.
In other words, it is definitely more than 2% of teachers that are called “ineffective,” but I don’t know the true cutoff.
If anyone knows the true cutoff, please tell me so I can compute anew the percentage of teachers that are arbitrarily being kept from tenure.
Categories: education, rant, statistics
1. April 2, 2015 at 7:33 am
They could instead simply award tenure on a random lottery, after negotiating with the union the number of new awards each year. This would be “fair”, right? (Maybe not fair to the students, but they never matter.) For better or worse, many programs are controlled in this way. For example, some public colleges explicitly select new students randomly from all qualifying students. Many colleges have a system which in effect is highly random, but not acknowledges to be so.
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2. April 2, 2015 at 9:03 am
You have as much chance of making headway among hoi polloi with this explanation as you will of conveying why Marilyn Vos Savant was right about the Monty Hall problem.
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• April 3, 2015 at 8:53 am
I hope you are right.
Most people who listen do come to understand that the right strategy in the Monty Hall problem is to switch and basically anyone who you can get to play the game a few times.
Let’s not be too pessimistic about or dismissive of the public.
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3. April 2, 2015 at 9:19 am
Is there actually a fixed cutoff that’s independent of the test results? And if so, who figures it out, and how? I mean, if there’s some good way of figuring out that some percentage of teachers are ineffective, then — drumroll — shouldn’t we just use that method to figure out who those teachers are, instead of using VAM? A good teacher evaluation system should produce a convincing estimate of the number of ineffective teachers as an output, not assume it as an input.
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• April 2, 2015 at 9:43 am
Right, I mean there has to be, if there are bins, which there are.
In some sense I think the VAM is just a complicated obstacle to hide the precise mechanism for this cutoff.
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4. April 2, 2015 at 11:31 am
Why is there even tenure? Its not like we’re protecting academic freedom at the grade school level (the only valid reason for it, the purported reason it exists at the university level). Few other professions have guaranteed emploment. If we didn’t have tenure we wouldn’t need all this other convoluted BS.
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• April 2, 2015 at 11:36 am
We have tenure because it makes the job more appealing to the people who are considering teaching as a career. It is not strictly necessary, but if we remove tenure and don’t improve the conditions in other ways, we will have fewer good teachers. Their alternatives will be more pleasant.
That’s already pretty much true given the current climate, of course.
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• April 2, 2015 at 2:55 pm
Academic freedom does sometimes matter in secondary education. There was an attempt to fire my high school American history teacher, ostensibly for teaching religion. I only knew about it because I was called in as a witness. It was a ridiculous charge, but without tenure he could have been fired without cause. From what I later heard, it had to do with bad feelings between his wife and someone in the school administration.
Also, remember the Scopes Monkey Trial. History and science are areas where a lot of people want children to learn mythology. Teachers need to be protected against that.
And, as mathbabe indicates, job security is a way to get teachers to accept lower pay than they would get otherwise.
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5. April 2, 2015 at 12:16 pm
“Think of it as a biased coin flip, and 30% of the time – for any teacher and for any year – it lands on “ineffective”, and 70% of the time it lands on “effective.” We will ignore the other categories because they don’t matter.
How about if you look over a four year period? To avoid getting any “ineffective” coin flips, you’d need to get “effective” every year, which would happen 0.70^4 = 24% of the time. ”
This is assuming independence of the results from year to year. Correlation actually creates a rigorous test for how well the program works. If for the nth year, P(Ineffective)(n) is approximately the same as P((Ineffective)(nthyear) (given that (ineffective)(n-1)), then the is independence, and the results are crap. If on the other hand, they are highly (and positively) correlated, the program would be at least somewhat vindicated.
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• April 2, 2015 at 12:17 pm
Yeah, no, it’s not. That’s the first link.
I mean, it’s not 0% correlated, but it’s not highly correlated.
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• April 2, 2015 at 12:25 pm
Possibly. Correlation in the same year for different grades might be different that correlation in different years for the same grade.
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• April 2, 2015 at 12:25 pm
Rubinstein studied both.
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6. April 2, 2015 at 12:42 pm
The NYS Commissioner of Education sets the cut-off.
The position of NYS Commissioner of Education is now vacant, after John B. King’s resignation at the end of 2014 to take a job as Secretary Arne Duncan’s chief advisor. http://www.nytimes.com/2014/12/11/nyregion/john-king-new-york-state-education-commissioner-is-leaving-for-federal-post.html
From the NYS budget bill
THE COMMISSIONER SHALL DETERMINE THE WEIGHTS AND SCORING RANGES FOR THE SUBCOMPONENT OR SUBCOMPONENTS OF THE STUDENT PERFORMANCE CATEGORY THAT SHALL RESULT IN A COMBINED CATEGORY RATING. THE COMMISSIONER SHALL ALSO SET PARAMETERS FOR APPROPRIATE TARGETS FOR STUDENT GROWTH FOR BOTH SUBCOMPONENTS, AND THE DEPARTMENT MUST AFFIRMATIVELY APPROVE AND SHALL HAVE THE AUTHORITY TO DISAPPROVE OR REQUIRE MODIFICATIONS OF DISTRICT PLANS THAT DO NOT SET APPROPRIATE GROWTH TARGETS, INCLUDING AFTER INITIAL APPROVAL. THE COMMISSIONER SHALL SET SUCH WEIGHTS AND PARAMETERS CONSISTENT WITH THE TERMS CONTAINED HEREIN.
Rubinstein’s analysis of the lack of year-to-year correlation of teachers’ scores in VAM:
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• April 2, 2015 at 1:07 pm
The “subcomponents” in the law are (1) mandatory tests and (2) optional tests that the state will develop or approve. There’s another “component” of evaluation, based on observations.
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7. April 2, 2015 at 11:40 pm
According to this document from NYSED released in December (full slides at http://www.regents.nysed.gov/meetings/2014/December2014/Evaluation.pdf, linked from press release at http://www.nysed.gov/news/2015/2014-preliminary-statewide-evaluation-results-released) the percentage of teachers rated “ineffective” on the growth measure itself has consistently been at around 6% for the last few years (slide 5).
So I think this is not the strongest avenue of attack against the use of VAM in teacher evaluations (especially given that so many others exist). Kirabo Jackson of Northwestern said (to EdWeek) about his own research on using VAM at high school: “We know there are other ways in which we could be spending our energy to improve student outcomes… my takeaway is that this [VAM] is not it.” I think his statement applies far more broadly.
More generally, I wonder if showing the weakness of the model is even the most salient problem with this approach. Why make a whole infrastructure predicated on identifying the 10% (or 5%, or whatever) of the “worst” teachers, when its effects on the other 90% (or 95%, etc.) of teachers are at best distracting, and at worst actively making this much larger group far LESS effective at promoting student learning?
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• April 3, 2015 at 6:55 am
Hi,
Thanks for the link! Actually I think the relevant numbers are on page 5 of these slides, where it breaks down each of the 4 categories. In our case we are looking at anyone who lands in category 3 OR 4, which is to say 16%, 17%, and 16% in the three years represented.
If you work that out, we’re talking about about 13% of teachers getting denied tenure purely by dint of this ridiculous coin flip. Given that there are around 70K school teachers in the system, only some fraction of them are untenured at a given time, but more than you might expect because the turnover is terrible. Let’s guesstimate that 5% of teachers are up for tenure in a given year, which still means 3,500 teachers. 13% of that is 455 teachers who have been excluded from tenure simply based on this random number generator alone.
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8. April 3, 2015 at 5:14 am
This is why it is unacceptable to have uneducated illiterates running educational systems. By “uneducated illiterates” I mean people who can’t explain what angular momentum is.
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9. April 3, 2015 at 7:34 am
Meanwhile, over here in Vermont, the State Board of Education recently put out a “Statement and Resolution on the Appropriate Use of SBAC Standardized Tests and School Accountability” (http://education.vermont.gov/documents/EDU-SBE-031715_SBAC_Resolution.pdf ). It’s worth reading the whole thing, but here’s a taste: “RESOLVED that until empirical studies confirm a sound relationship between performance on the SBAC and critical and valued life outcomes (“college and career – ready”), test results should not be used to make normative and consequential judgments about schools and students…”
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10. April 4, 2015 at 10:00 am
Thanks so much for continuing to care and to write about this. Here’s a link to a New Yorker article describing the atmosphere and the actions encouraged by reliance on test scores as the only data on which teachers and schools are evaluated:
And let us spend a moment thinking about the God-forsaken school personnel in Atlanta who have been prosecuted under RICO statutes and face not just losing their jobs and paying fines, but serious jail time. Did these teachers and administrators just suddenly decide one day to manipulate scores, or was the pressure to do so overwhelming. . .
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• April 4, 2015 at 10:07 am
Linda,
Yes, thanks for tying this to that ridiculous sentencing. The worst is how the reporting sometimes actually makes it seem like these teachers are the problem with the system.
Cathy
On Sat, Apr 4, 2015 at 10:00 AM, mathbabe wrote:
>
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11. April 7, 2015 at 1:48 am
I read this quickly so I may be wrong, but you are assuming independence in your calculations, which seems wrong. If a teacher got an effective eval in the first year, the conditional prob, of an effective eval for the second year should not be 0.70 any more, it should be much higher.
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• April 7, 2015 at 6:16 am
It’s true, I’m assuming independence. In fact I think there’s a modest year-to-year correlation, somewhere around 20%. I think, if I took this into account, my measurements would go up a bit, and we’d be getting rid of even more teachers for dubious reasons. If you want to know why the correlation is so low, please read the original post, which links to Gary Rubinstein’s work. That low correlation is the basis of much of my outrage.
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1. April 2, 2015 at 10:07 pm
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2017-03-27 14:35:41
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https://forum.nationstates.net/viewtopic.php?p=37703246
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NATION
## World Cup 86: Roleplay Thread
A battle ground for the sportsmen and women of nations worldwide. [In character]
Trolleborg
Chargé d'Affaires
Posts: 475
Founded: Oct 11, 2010
Inoffensive Centrist Democracy
Greetings, compatriots!
It’s we, TTV from the Stadium of 1st December. Our team play last group game of WCQ86 against Kita-Honde.
We can say without exaggeration, that the whole country celebrated the victory in the penultimate game, which opened the way to the World Championship for our lads. Undoubtedly, the team would be met by a huge crowd, and no persuasion of the highest level, starting with the national leader, would have helped avoid it, but the captain and head coach of the team made an appeal not to arrange celebrations until the end of the tournament, as the one game still ahead. In it, our guys had to prove who they were face-to-face with one of the grandees of world football, and try to confirm that latest string of victories were not a chain of accidents and "the luck of fools". Kita-Honde, after a sudden defeat from Fluvannia on its own turf, as we all worry, wants to recoup on upstarts, and we all have in mind 0-3 defeat in the first game. Three points give them ticket right into the world cup, almost any other result condemned team to the excruciationg pain of the playoffs.
TROLLEBORG – KITA-HONDE 2:0.
Our team, as it turned out, was not dizzy with success. Lads entered the field with full composure, play very carefully and at the same time with great dedication, not allowing opponents to do as they are able in their best moments. The first real moment near the Kita-Honde goal was created already in the 3rd minute, but Kerr missed the net from several meters - however, his position was really difficult. Then the whole bunch of our players had the opportunity to open score - notably Valgren, Gulbranden, again Kerr, and it was Gulbranden who was lucky at the end. He successfully took advantage of a long and accurate pass from the flank and accurately put his head where needed to score.
The success of our guys did not lead them apart from the way they take, they continued to play as they played, however, until the break there were no serious opportunities to change the score.
In the second half, the game was all about the same, and it seems that the guests were very seriously influenced by the setback in the last game, they did not seem emotionally to move away from such an unexpected defeat, they did not make a real moments at the gate of Kotsson, but at the opposite the gate dangerous situations constantly developed. We even saw Troger come to serve a head shot after the corner, which he hasn't done in international games for a very long time.
Towards the end of the game, the referee punished the guests with a penalty kick, which Kerr converted calmly,
and, perhaps, in the last minutes our guys could have brought the score to a devastating one, having equalized the balance with Kita-Honde, but could not do it. Nevertheless, 2:0 is a victory, and the victory is not accidental, really worthy and stylish.
Our guys deservedly ran a circle of honor around the stadium, and then millions of Trolleborg City residents and billions of country citizens everywhere enjoyed a solemn fireworks display in the honor of our national team.
TTV congratulates all football fans and everyone who cares on winning the qualifying tournament group. We hope that it was interesting for you to watch our broadcasts and we hope that you will continue to stay with us.
Finally, we remind you that the national leader will go on the air with a speech dedicated to our players tomorrow.
Last edited by Trolleborg on Fri Sep 25, 2020 6:43 am, edited 1 time in total.
Terre Septentrionale
Diplomat
Posts: 591
Founded: May 31, 2019
Ex-Nation
Terre Septentrionale easily beats Squornshelous in first leg
Vogsphere, Squornshelous - It was really unexpected, Terre Septentrionale won a playoffs game 6-2 against Squornshelous, who are participating in their 40th qualifications. This is our biggest win, tied with a 6-2 win against Lashnakia in the BoF 70, a 6-2 win against Unimon in the 86th World Cup qualifications and a 7-3 win against Hwiccemark in the same qualifications. Luckily Squornshelous granted us our visas in time plus a few hundreds more for our fans and players' family.
Terre Septentrionale attacked at the 5th minute when Yannick Ksiezniakiewicz passed to Sandrine Fontaine who dribbled with the ball and passed to Sandie McCracken who kicks it and it's a goal! Squornshelous tied the game at the 17th minute when Lucia Krupin made a cross to Tove Anker who kicks it in the net and the game is tied at 1-1. Sandie McCracken was dribbling with the ball at the 22nd minute and she kicks the ball when Iosif Chaika tackled her after she shot. Chaika got a red card for his late tackle, so his team will play with 10 men and women for the rest of the game, and the Nordiques gets a free kick. McCracken has to be carted off the field and Zvonimir Kordic replaced her. Roksana Kolodziejczyk takes the free kick, she shoots and she scores! 2-1 Terre Septentrionale. The Reprezentacja Nordycki added another goal at the 30th minute, on a corner kick when Élisabeth Normandin kicked the ball in the penalty area and Yannick Ksiezniakiewicz headed the ball into the net. 3-1. Terre Septentrionale attacked at the 41st minute when Roksana Kolodziejczyk dribbled with the ball and passed to Zvominir Kordic who made a cross to Fryderyka Cwierczakiewiczowa. Fryderyka Cwierczakiewiczowa kicks the ball and it's in, 4-1 Northmen and this is the score at the halftime.
Fryderyka Cwierczakiewiczowa scored her 2nd goal of the game at the 57th minute when Wlodzimierz Waclawczyk made her a short pass and she kicks the ball into the net. the score is now 5-1. Squornshelous answered back immediately at the 61st minute when Natalia Berglund dribbled with the ball and passed it to Havard Popov who kicks it and he scores. Terre Septentrionale's lead is reduced to 5-2. the Northmen scored again at the 79th minute when Przemyslaw Tkocz, who subbed in at the 69th minute, dribbled with the ball and passed to Roksana Kolodziejczyk who kicks the ball and it's in. She also scored her 2nd goal of the game and the score is now at 6-2 for Terre Septerntrionale. This is the final score.
BOXSCORE
5th minute: Sandie McCracken assisted by Sandrine Fontaine
17th minute: Tove Anker assisted by Lucia Krupin
22nd minute: Roksana Kolodziejczyk (free kick)
30th minute: Yannick Ksiezniakiewicz (corner kick)
41st minute: Fryderyka Cwierczakiewiczowa assisted by Zvonimir Kordic
57th minute: Fryderyka Cwierczakiewiczowa assisted by Wlodzimierz Waclawczyk
61st minute: Havard Popov assisted by Natalia Berglund
79th minute: Roksana Kolodziejczyk assisted by Przemyslaw Tkocz
YELLOW CARDS
12th minute: Sandrine Fontaine
34th minute: Slawomir Skrzeszewski
69th minute: Ralf Tjaeder
RED CARDS
22nd minute: Iosif Chaika
2nd leg
The 2nd leg of this playoffs series will be played at Stadion Hydro-Polnocna in Trzebiegoszcz, Township of Pomerania. Iosif Chaika will likely be suspended for this game because of his red card, unless Squornshelous decides to appeal his suspension. Sandie McCracken will be out for a couple of weeks and Zvonimir Kordic will replace her in the starting XI.
Starting XI: Vainqueur - Niedzwiecki - Skrzeszewski - Brzezanczyk - Normandin - Ksiezniakiewicz - Fontaine - Kolodziejczyk - Waclawczyk - Cwierczakiewiczowa - Kordic
Wlodzimierz Pietruszewski,
Senior writer,
La tribune Septentrionale
Nation name: République de Terre Septentrionale | Trigramme: RTS | Capital: Ville Jacques-Cartier | Maps
Ranks: Hockey: 20th | American Football: 7th | Baseball: 17th | Association Football: 23rd | Rugby Union: 21st
Champions:
Runner Up: Cup of Harmony 76, International Baseball Slam XI
3rd Place: World Volleyball Expo X, International Baseball Slam XII, World Lacrosse Championship XXXV
4th Place: Arena Bowl VI
World Cup participations: WC 85 (3rd place in group), WC 86 (3rd place in group)
Busoga Islands
Spokesperson
Posts: 192
Founded: May 31, 2018
Corrupt Dictatorship
The Busoga Chronicle
The Islands #1 News Source!
BISA announces they will stick with O'Donnell despite 'dismal' World Cup Qualifying effort
Jay Khano runs for the ball during World Cup 86 Qualifying
LAKISKA, SOUTH ISLAND- To say that World Cup 86 Qualifying was a disappointment for the Busoga Islanders would be an understatement. THe only way to describe this cycle would be to say it was an 'absolute, outright, and complete failure', in the words of BISA(Busoga Islander Soccer Association) officials. There were hopes for the Islanders heading into the campaign. While the group was a tough one, our performances in World Cup 85 Qualifying offered hope. We were on the edge of the Finals, although we could not make it, and we won our Cup of Harmony group for the first time ever. We were ranked as high as #45 in the KPB rankings for these Qualifiers, and in Pot 3, there was certainly a chance for us to make a run at 2nd place and make the playoffs.
Unfortunately, that did not happen. O'Donnell ran with largely the same squad that was in World Cup 85 Qualifying- just now a few years older. From Pot 3, we fell down four places, all the way to 7th. We only earned 19 points across 10 games, a terrible performance for a team ranked as highly as we were, in the top 50. And it is seen in our fall in the rankings- we fell a stunning 17 places, from 45th in the KPB to 62nd- from a strong Pot 3 side to a mid-level Pot 4 side, in terms of World Cup Qualifying. Where we were hoping to maybe move into the top 40, and keep this project moving forward, we instead took a large step backwards.
Bethany O'Donnell, the Brenecian manager, did not mince words. "This is not what we were looking for- this campaign was a disappointment." She told reporters after the campaign dismally, with a 3-0 home loss to an already eliminated Mercedini. "The standard is held higher here. I understand why fans are disappointed. They were expecting much more than this, and we are capable of much more than we gave over the course of the last 18 games." Of course, with the huge fall in the KPB, one might expect that she might be fired. Somewhere else, maybe. But BISA's President has committed to her overseeing the project.
Here is what he told the media. "Obviously, we aren't happy with this performance- nobody is. But we have to say this- do we want to start over, again, or are we capable of improving upon this? I believe that we are. This is a bump in the road, but everyone in sports has to overcome adversity before they can reach the place that they ultimately want to go. We believe in O'Donnell- the performances in the last World Cup Qualifying campaign, and in the last Cup of Harmony, specifically, make us believe that she is the right person to continue to lead us."
Now, there's a lot to unpack there. But let's skip all that- his statement is best understood by what he didn't say. What somebody in his position knows is absolutely true, but couldn't say. We aren't BISA executives, however, so we'll say it- managing the Busoga Islands is one of the toughest jobs in international football. Why? This is a small nation. We are a nation of 9 million people. A small country, where cricket is equally popular as soccer. We only have six professional clubs in the country- and there is no locally controlled domestic league, they all compete in Banija. Now, of course, that's beneficial- the Banijan Soccer League is the 12th best soccer league in the sport, according to the IFCF ranks, and they got the money to match. Club football would be much, much worse if those six clubs were on their own.
Another thing that makes this difficult is, of course, the inevitable comparisons to Banija. We are a small fish in a big pond. The Glorious Southwest is a difficult place to play. Siovanija & Teusland is rising quickly up the ranks, and is set to make the jump from 'good' to 'elite'. Valanora has won the World Cup 5 times. Both Equestria and Farfadillis have won in recent years, with Equestria winning World Cup 83 and Farfadillis winning 84. And of course, the nation where our clubs compete, Banija, is ranked #1 and just went 18-0-0 in World Cup Qualifying. Somebody, please beat them in the World Cup- the last thing this world needs is for the Banijans to get their hands on a trophy. But with our nation's history as a former province of Banija, and with our domestic clubs competing over there, it's the natural comparison.
When you combine expectations without resources, it makes for a lot of unhappy fans. Plus, this is the type of job that needs stability. Those big nations will always get huge managers. But how attractive is this job? On it's own, who knows? But when you combine that there is talent here, and at least some resources, you can make it attractive if you offer something else- stability. If we're a small country that is known to be quick to fire its managers, we're doomed. O'Donnell is committed- she is at an age where she isn't using this as a launching pad into another big job(we think, anyways). But that commitment back to her can pay dividends in the long run, even if this was a terrible Qualifying campaign.
All of that is true. And are legitimate reasons why you would continue to commit to O'Donnell. We'll see what happens in the Cup of Harmony(if we make it- the oddsmakers have us at about a 50/50 shot to get into the tournament). But, of course, then you get to the rumor mill. There are big changes coming in club football in Banija- but of course, we do not know what. The Banijan Soccer League has announced that they'll let their media rights deals expire at the end of the season. Now, of course, that is something that is rarely seen- usually, these deals are announced one year before the old ones expire. But letting them expire implies change is coming, a big enough change, we assume, to positively influence media rights deals, to the point where the league is willing to start those negotiations mere months before the next season begins.
What changes are those? Structural? We have no idea. Technically, BISA has no formal influence in the RBSA(Royal Banijan Soccer Association). The six Busogan clubs are full members of the RBSA, of course, but not BISA directly. They have no direct vote on changes in the RBSA's domestic structure- only the 68 club owners do. Naturally, however, there is some cooperation between BISA and the RBSA. But rumor has it BISA is fighting hard for the Busogan clubs. Fighting hard for what, exactly? We have no idea. But if big enough change is coming to the domestic football structure in Banija & Busoga, BISA may not be in a position to conduct a coaching search for the national team. Admittedly, we're spitballing here, but maybe they think that they need to focus their energy on the club side right now, and leaving the veteran manager in charge is best while they figure out what the heck is happening in Banija.
So, long story short- Bethany O'Donnell will remain in charge for the foreseeable future. She will certainly coach at the Cup of Harmony, if we make it, and at AOCAF LXII. There are all sorts of possible reasons this will be happening, none of which will be said out loud by her bosses. Expect her to change things up for the Cup of Harmony, and to bring in a number of new faces- lack of squad competition probably did not help the squad this time around. Fear not- they will do what they can to put that disastrous Qualifying campaign behind them.
Kandorith
Posts: 1562
Founded: Aug 26, 2009
Shinonome Kyoai - International
News From The Empire
Difficult Away Match For Kandorith
South Covello 0–0 Kandorith
As expected the match against South Covello would be a difficult one, the thing which wasn't expected is that it would be this difficult. A very rough and disorientating match full of errors and irritations was the story of the night and with that came a very agitated Hayabusa as he was warned by the fourth official.
There was an atmosphere of hostility towards the Kandorese fans and the Kandorese team as the match started which did not improve throughout the match. Hayabusa and the team however, would not concede to the verbal home side of South Covello. The match did suffer from it eventually; as Kandorith failed to put a ball near the net for almost the entire match.
The second half was not much better for the Lotus Warriors as the ultra-defense minded South Covello team tried to completely shut down the match but could not find any movement through the Kandorese midfield. Where Kandorith finally got more shots on goal and even on target, it started to look more hopeful. Hikoshige Matsushita launched an attack in the 70' minute and rocketed the ball towards goal where Katano was awaiting the lobbed pass. As Katano was shoved down by the South Covello defence the whistle sounded, surely it would be a penalty for Kandorith. But no, the penalty and appeals were denied and Masada Katano was shown the yellow for unknown reasons.
Hayabusa absolutely lost his temper after this yellow card as it was the second goal chance which was taken away from the Kandorese. Earlier in the match Masada Katano was ruled as off-side while he was clearly onside and the officials refused to hear any appeal and handed the captain, Tanaka, a yellow card for arguing with the referee after the debacle. Hayabusa yelled at the officials to take a look at the obvious foul against a goal scoring chance but it fell on deaf ears as he was ordered to calm down or else he would be removed from the field.
The match lasted until the 93' minute without any noteworthy events as Kandorith tried to launch one last attack before the 95' minute. All Kandorese players poured towards the South Covello goal and left only two defenders to go for an all-or-nothing attack but the South Covello defense would not budge. The last moment of the match saw an amazing shot from Hisakane Hayashi which hit the bar, but as the ball rebound and fell perfectly at the feet of Katano once again; he was ruled offside once again. In a final outburst Hayabusa was shown a red card directly as the referee no longer wished him on the field. The final whistle sounded as Hayabusa took a shot at an innocent water bottle with his right foot and the match ended in a 0 - 0 draw where for Kandorith it does feel as a defeat. Hayabusa later did not wish to comment on the sending off but ensured that the home-side advantage will be the match of the decade for the Kandorese team.
"We should have done better or found more holes in there defense that I must admit" Hayabusa said. "But eventually it wasn't just a match against South Covello but also the referee. I truly feel we were wronged a few times and should have had at least one goal. I will not say too much about it as I am in no position to do that and of course the match has been played and ended in a draw.
"I believe it should have been a win for us, but we will absolutely do better. Next we will meet them in one of the most beautiful stadiums in the world and this is our home side. In our stadium we will show them what Kandorese football is about and this is where we will take the win. And I promise to behave towards the officials who will be leading that match" Hayabusa said as he stood up and chuckled.
The second leg of the playoff against South Covello will be one of the most crucial matches for Kandorith in recent memory as this will be the first chance to get to the World Cup for a long time. The fans are of course still hopeful and in the polls on social media the Kandorese are the clear favourite with a 90% vote for Kandorith winning, 9% of the match ending in a draw where Kandorith would win and eventually a 1% vote for a South Covello win. As the capital prepares for the match, the nation will pray and be ready to cheer on their team across the entire nation.
Great Empire of Kandorith | 大宮来国 | Dai Kanyori Teikoku
| Overview | Constitution | Anthem | Imperial Anthem | Armed Forces | Foreign Affairs | Emperor
Shinonome Kyoai Headlines:
BREAKING NEWS: Hiyashi Corporation to unveil strategy for the coming year. ● Winter celebrations in full swing throughout the nation, celebrating the ending of the year and asking the gods for a prosperous 2021.
Chromatika
Posts: 1363
Founded: Aug 05, 2015
Democratic Socialists
Part 1: The Barbed Inn, Chromia
Part 2: The Emerald, Ming
Part 3: Killjoy, Cortez River
Part 4: Jordan's Talk
Part 5: Islander Dome, Myana
Part 6: Busukuma/Mid-Qual Thoughts
Part 7: Triple B, Deprí Sanar
Part 8: The Date, Alnio/Chromia
Part 10: Before the Last Day
Part 11: TJUN-ia Preview
World Cup 86 Qualifying Playoff, First Leg:
Score: TJUN-ia 1-2 Chromatika @ National Stadium, TJUN-ia City
Goals: Mora '39, Victoriane '82
Starting XI: Tioux; Xiao - de Aea - Ken - Ilya; Victoriane - Armageddon (C) - Dragana; Mora; Andisori, K. - Vidal.
Substitutes: Will -> de Aea ('54), Dias -> Armageddon ('62), Kuzami -> Mora ('71)
NATIONAL STADIUM, TJUN-IA CITY, JB TJUN-IA -
A late tap-in from Kaytlyn Victoriane turned out to be the difference as Chromatika edged out TJUN-ia to take a narrow win, on the road, before heading back to their home confines in the Capitalizt Dome.
Vladimir Podolov opened up the scoring in the eighteenth minute when a Pedro Vasquez cross beat out Kai Xiao's positioning, leading to a one-on-one with Ferret Tioux.
Hermaeus Mora equalized with a free kick that snaked in to the upper right corner past the outstretched hands of Harold Gylfisson.
The game remained close after that with both squads having decent chances until a late counterattack by Kuzami led to the tap-in chance that Kaytlyn Victoriane completed.
The result gives Chromatika a definitive lead as the series moves to Chromia, as the two away goals will mean that if TJUN-ia were to win 1-0, Chromatika will still win via away goals.
"We are fortunate enough to be in this situation," stated Victoriane after the match, "TJUN-ia fought well and fought hard. They're definitely not a team to be taken lightly."
Giles Ken proved why he was starting alongside Henri de Aea by clearing four headers and providing many tackles in the Chromatik backfield.
As Richard Valens and Tim Landers are so close to qualification. Will the team make it?
Team Selection
PO2: vs. TJUN-ia
Starting XI: Tioux; Xiao - de Aea - Ken - Ilya; Victoriane - Armageddon (C) - Conrad; Mora; Andisori, K. - Vidal.
Reserves: Williams, Chapman, Anderson, Will, Austic, Thorben, Dias, Dragana, Kuznetsov, Kuzami, Sierra, Andisori, B.
Former User of the Nations of Yesopalitha and Falconfar
Regional Tournaments: 52 (2nd), 53 (4th), 54 (QF), 55 (Champions), 56 (Ro16), 59 (Ro16), 60 (QF), 61 (4th), 62 (Ro16), 63 (2nd)
WC Proper Appearances: 75 (Ro16), 76 (SF), 77, 78 (QF), 79 (Ro16), 80 (Ro16), 81, 83, 84, 86, 87 (Ro16)
CoH Appearances: 77 (Ro16), 85 (Ro16)
KPB Ranking: 15 (Post 87)
RP Population: 20 million
Silver Commonwealth
Posts: 1822
Founded: Aug 16, 2018
Psychotic Dictatorship
(This is my last RP post ICly and basically a ''farewell'' one, so it doesn't count towards the score obviously)
Enrico was reading a bunch of papers with a concerned look in an office room of Salgado Stadium, and while he had called a meeting with all of SC's team's players (Both reserves and the ''main'' team) assistant coaches and even the manager Jesse Camacho, not much words had been spoken yet.
''What is wrong? You seem to be quite concerned, Enrico.'' Rafael Kanashiro was one of the first to break the silence - the third goalkeeper of SC's team, and the only on who hadn't been on the field at all during the World Cup, as Ximenes was chosen instead.
''You should already be able to tell what is going on by now, as we have been in plenty tournaments when awaiting for results.'' Enrico responded in a brash tone. ''Anyways, as it has become pretty obvious now, we didn't make it through the qualifications, and ended up getting eliminated from the world cup. What a great story to tell when we arrive back home!'' Enrico finished sarcastically.
''So, you can basically pack up your sports bags and put your SC's team uniforms in the closet because there is nothing else left for us to do than to piss off from this event.'' Jesse Camacho added to Enrico's words. Initially footballers didn't react much in words, although it could be seen that the results were devastating or at least unpleasant for them.
''But wait, what was our score?'' The defender Zanetti was the first one from footballers to speak up. ''Don't tell me that we are at the very bottom and didn't have at least some successes!''
''Well, we are in the 8th place on our Group's scoreboard, and I can imagine that the situation wouldn't look too better if we compared it to the rest of the nations in tournament, as we would fall in the category of nations which didn't perform so well. You can already guess that we had 10 loses, and that ain't looking too good right now''.
Then the team's captain Gabriel Vila had something to say. ''That essentially means that we're screwed when return home to our clubs and interviews - I can imagine that some of football thugs will kick our face to purple color!''
''Easy, easy.'' Enrico tried to speak calmly. ''As far as I know, the competition was hard from the beginning, and from a single group only the two teams at the top qualified, and in some it will be only one due of the second places in groups being in two play-off phases from which even more nations will be cast away, and I can imagine that for them the fan disappointment back at home will be even greater. So, I say that we actually did pretty good for the first time. Don't let your heads down - I am sure that most of people back at home will understand.''
After this attempt from Enrico to calm down his team there were lots of other conversations, with some of them being a chit-chat between the footballers, while others were discussing serious topics in which the manager and coaches would participate as well. ''So, this means that we can finally go back to home and rest/play against regional teams, then?'' One of the defenders from the ''main'' team (Eric Soares) could be heard in the middle of the loud conversations.
''Hold on, we aren't done yet.'' Enrico stopped Soares. ''There is still one task, or tasks rather, left for us - those are interviews to our media. Even with technically a single state channel there are countless of branches about various topics (Which isn't that much smaller in terms of numbers if compared to IRL media channels), and there might be some foreign reporters as well, so it will probably take quite a lot of time''.
Some time later the SC's team and Enrico were ready to be interviewed by SNN's reporters, but Enrico still had that eerily pressure of dread as he walked closer towards the main room with media microphones and cameras, as if he would be afraid of his future or even life. He was the first one to arrive, and his team would arrive soon after. While some of the footballers had got significant fame back at home, thus making several reporters to try seeking an interview from them, most of the SNN's reporters were focused on Enrico as he was the head of team and responsible for the training, preparations, and so on.
However, it soon got pretty chaotic as it was hard to make out what was being said as multiple reporters spoke at the same time, and there was crowding as well, due of journalists trying to get closer to Enrico.
''Calm down!'' Enrico spoke in a harsher tone. ''Speak one by one so I can understand your questions and answer them.''
The first reporter to ask Enrico a question was from SNN Brazil - a slightly tall woman with a suit from 1930s and brown eyes. It is likely that Enrico chose her to be first due of him being from SC's Brazil himself.
''We will start with the question which perhaps is the most popular one and to which people want to get an answer: How did your team perform in the Qualifications, and did it manage to pass?'' The reporter asked.
''Well, uhm...'' Enrico started slowly. ''Thing is, that our team...'' (Further interview and questions from other SNN's reporters omitted)
*** A few days later in one of SNN sports articles ***
SNN Sports
Our team doesn't make it through the qualifications, and thus gets eliminated from the World Cup 86!
''In the press conference Enrico announced that SC's team didn't make it through the qualifications, and thus was eliminated from the World Cup. It means that the international team will be assembled again only when it would apply for a new tournament, and until then the footballers would return back to their clubs, or to their regional team in case if they would be assembled for SC's regional tournament. Head coach Enrico, assistant coaches and the rest of the team would continue to give answers to questions of various reporters of SNN's mass media branches such as SNN Brazil, SNN South America, SNN Europe, etc, before leaving the World Cup 86 and the qualifications behind, thus finishing the chapter in a story. While obviously fans were disheartened about team not qualifying, most of them deep in heart would understand that such an outcome wasn't unexpected as SC's team would face a lot of strong foreign opponents in a setting it hadn't experienced before, albeit there were still some which were problematic in the streets and the cops still worked in heightened caution for days after the tournament until the heightened attention of football would die down, and it would be easier to control dangerous elements in regional matches instead. However, the outcome of this endeavor still sparked lots of debate within SC's football community about certain way of things in how SCFA has ran so far, especially about mixed teams - SCFA recently released a statement in which it was said that ''Sending a mixed team based on player skill is considered in future, and it is likely that some players from women's football teams could receive a ticket to World Cup tournaments''. While receiving condemnation from more traditionalist elements, some were relieved that ''multiverse had taught us a thing or two''.''
''In the Group 6 (Also called as ''Toilet Group''), SC ended up in 8th place, only below Sylestone and Southwest Eastnorth, and while it isn't a particularly great spot, readers should keep in mind that it was the first time when our players participated in such an event. SC had 18 points overall, 3 draws, 5 wins, and 10 losses, along with 27 GF, 44 GA, and GD in negatives: -17. Only country which automatically qualified from 6th group was The Holy Empire with only a single draw and loss, and 49 points, while Audioslavia got to playoffs with 38 points overall, 12 wins, 2 draws, and 4 losses. As we can see, there are 24 countries which already have automatically qualified, and 12 which still have a fight ahead for their spot in next stage of World Cup through playoffs. After doing quick calculations, it seems that only a small amount will squeeze through the last trial of fire by playoffs. The list of countries from all tournament groups which qualified or are in playoffs are below.(In brackets there is the number of the group)
Qualified: Brenecia (1), Hampton Island (1), Ko-oren (2), Pasarga (3), Eura (3), Nephara (4), Siovanija and Teusland (5), Turori (5), The Holy Empire (6), Banija (7), Vilitia (8), The Sarian (8), Valanora (9), Reçueçn (10), Mriin (11), Trolleborg (12), Kelssek (13), Equestria (13), Farfadillis (14), Cassadaigua (15), Starblaydia (16), Zwangzug (17), Baker Park (18), Savojarna (18).
Playoffs: 95X (2), Terre Septentrionale (4), Audioslavia (6), Chromatika (7), Kandorith (9), Squornshelous (10), TJUN-ia (11), The Hinodejin Empire (12), Graintfjall (14), Quebec (15), Darmen (16), South Covello (17).''
''It is unknown if SC's international team of Association Football will participate in future World Cup events, because while SCFA has considered such a possibility, thing is that the funds for sport also aren't limitless, and other sports associations have to receive their part too. Also, not all of SC's regional ministers and people are fine with spending money for the sports tournament, and would rather prefer seeing it go towards rebuilding the infrastructure and investing in security. ''They throw the money for preparing their team for a sports event such as world cup, while it could have instead gone for funding the introduction of modern food industry in the poorer regions, especially considering that some of them still use ration system due of the lack of supplied products.'' A politician from SC's Peru-Bolivia remarks in a meeting. Also, most of the players of SC's international team have went back to their Brazilian clubs which are preparing for regional tournaments instead. So, right now one can only speculate when, or if at all SC's international team will be assembled together again for an another try in the World Cup tournament of multiverse. But, even if they didn't pass through the qualifications, so didn't a lot of other teams, as qualifications were a tough business. And in the end they are still our team!''
✥ ᴛʜᴇ ɴᴇᴡꜱ ✥
- ꜱɴɴ
- ᴀʀᴄʜɪᴠᴇ
✉ ʀᴀɴᴅᴏᴍ ✉
- ᴀᴅᴍɪɴɪꜱᴛʀᴀᴛɪᴏɴ
- ꜱᴛᴏʀɪᴇꜱ
⚒ ᴛʜᴇ ʟᴇɢɪᴏɴᴀʀʏ ᴡᴏʀʟᴅ ᴏʀᴅᴇʀ ᴏꜰ ꜱɪʟᴠᴇʀ ᴄᴏᴍᴍᴏɴᴡᴇᴀʟᴛʜ ⚒
|
⚖ ᴀꜱ ᴛʜᴇ ᴍᴏᴅᴇʀᴀᴛᴇꜱ ᴀʀᴏᴜɴᴅ ᴛʜᴇᴍ ꜰᴇʟʟ,
ʀᴀᴅɪᴄᴀʟɪᴢᴀᴛɪᴏɴ ᴏꜰ ꜱᴄ'ꜱ ᴅᴇᴍᴏᴄʀᴀᴄʏ ꜱᴜᴘᴘᴏʀᴛᴇʀꜱ ʙᴇᴄᴀᴍᴇ ᴀ ᴍᴀᴛᴛᴇʀ ᴏꜰ ꜱᴜʀᴠɪᴠᴀʟ ☠
_[][][][][][][L'''][Σ][][~][][][]_
̿̿ ̿̿ ̿̿ ̿'̿'\̵͇̿̿\(▀_(▀_(▀_(▀_(▀_(▀_(▀_▀)_▀)_▀)_▀)_▀)_▀)_▀)/̵͇̿̿/’̿’̿ ̿ ̿̿ ̿̿ ̿̿
IIIIIIIIIIIIIII
Progress report: Despotist Free Territory's overview - slow, at a slug speed
cursed.socialism
''My hobbies? My hobbies are those at which I am bad at!'''
How based is this?
''I thought that I was a conservative. Turns out, I was just sentimental at times.''
Just Moi
Audioslavia
Game Moderator
Posts: 2939
Founded: Antiquity
Civil Rights Lovefest
Jeremy Jaffacake gave his phone the smuggest, most self satisfied punchable grin he could muster. And this was Jeremy Jaffacake. It was really quite smug.
"Yeah, Squornshelous" he said, audibly if to himself. "It's not nice being unexpectedly hammered by four goals at home, is it?"
"Jesus, Jeremy" came a voice from across the room. Daphnaes Hanson, known to most by his stage-name Defecating Daffodil, stopped gaffa-taping cables to the ground and looked over his glasses at Jeremy. "Are you still harbouring a grudge about something that happened in World Cup 5?
"Yes" said Jeremy. "No point in harbouring a grudge for a temporary amount of time. I mean it's not even a grudge if it ends, is it? Then it's just a strop"
"That doesn't make any sense" said Daphnaes
"Of course it makes sense" said Jeremy. "What's the point in being the kind of person who holds grudges if people find out that, after a certain amount of time, you tend to stop caring. Being the kind of guy who will plant and nurture a grudge forever kind of makes people, y'know, not want to mess with you."
"Maybe you should hold grudges at all?"
"Maybe you should be less ugly?"
"I'll remember you said that"
"Good. You're learning"
"6-2 though" said Daphnaes. "Squornshelous are seven kinds of screwed"
"2-6" said Jeremy. "And Squornshelous are seven sacksworth of shit in a six-shitsack sack"
"Bet you can't say that fast" said Daphnaes. Jeremy took a deep breath.
"Seven-satchmo-sassy-sit-a-seven-six-seven-sack-sack" said Jeremy
"Nailed it in one take, well done" said Daphnaes, going back to his gaffa-taping.
"How long are you gonna be? We need to get recording" said Jeremy
"Two minutes. One minute if you get off your butt and help"
"Just leave it" said Jeremy. "Just let the cables scatter across the floor. It's not as if anyone's going to trip, fall and hit their head and forget everything"
"You did that twenty minutes ago" said Daphnaes, bringing a frown out of Jeremy
"...no I didn't"
"Yeah you did. You shouted at me to 'get those cables put away you yellow fuck'"
"No I didn't" repeated Jeremy. "No. Twenty minutes ago I was sat in the corner over there, telling that canary he was a pillock"
"Exactly" said Daphnaes. "That wasn't a canary. That was me"
Jeremy thought for a moment.
"So. Running order for today's podcast" said Daphnaes, patting the last cable down. "We'll start with Squornshelous - you can do your anti-Squorn shtick - and then we'll look at the Audioslavia game. Anything particular you're going to want to say about the match?"
"Only that we could send the Audioslavia under 13 girls team out there to play the second leg and still hold Darmen to a draw. They're about as penetrating as..."
"Do not finish that sentence" said Daphnaes. "And... and are you sure you want to tempt fate like that? We all remember Abanhfl...
"We don't talk about that playoff" said Jeremy, quickly.
“The playoff where you hired an aeroplane to drop pamphlets all over Abanhfleft giving them a guide to the country that was hosting the Cup of Harmony that year, and then you lost on away goals and had to go to the Cup of Harmony yourself and then you told yourselves you were going to win the CoH and become the third team to get the Grand Slam but then you lost and ended the cycle with nothing?”
“Yeah that playoff” said Jeremy
“Thought so” said Daphnaes, sitting in his seat at the opposite end of a small desk to Jeremy. Two large microphones all but obscured his view of the Audioslavian. “I’m ready to start if you are”
“Aye” said Jeremy. “No mentioning Abanhfleft on the podcast”
“Of course” said Daphnaes, with a grin. “Of course…”
<uptempo dutch electronic music plays>
“It’s playoff time, and it’s Idiot Podcast time as we analyse Wednesday’s World Cup playoffs and look ahead to this Sunday’s return matches. Will there be no fury like a team Squorned as the ‘Psychychoes’ look to avenge their midweek terra-fying, and we analyse Audioslavia’s crucial away win over Darmen, but dar-many of you remember what happened just five cycles ago, when Audioslavia lost to Abanhflef…”
<two seconds of silence>
”Do you find your skeleton just goes un-used for most of the year? Well consider your lazy bones problem solved with the new app ‘Skullshare’, where you can rent your skeleton out to…
Last edited by Audioslavia on Thu Sep 24, 2020 5:32 am, edited 2 times in total.
TJUN-ia
Diplomat
Posts: 822
Founded: Oct 04, 2019
Civil Rights Lovefest
### Boot It!: Playoff 1st Leg - "Home Struggles Were Expected"
By Peter Davis
You hear that? That's the sound of Playoff Football in TJUN-ia City. The National Stadium had already seen 3 contests in the 18 previous games but now, it hosted the greatest of them all: Game 19. TJUN-ia was in the playoffs after drawing with Tikariot on the road, adding more success to an already successful campaign for David Seems and his Jaguars. A spot in the upcoming Cup of Harmony in Sylestone and Ethane was practically secured already, but the prospect of reaching the World Cup itself is something no mortal can refuse to fight for. And so The Anomalies of Chromatika, ranked 36th in the WCC before the campaign began, came to the JagBase for the first leg of this playoff fixture.
David Seems and his counterpart in Krytenian Richard Valens knew this match would be critical for the entire series, as TJUN-ia's woeful home form would take on Chromatika's woeful road form. With both team's strengths focused on the 2nd leg in the recently-renamed Chromia, any advantage taken here would set the tone for the game at the Capitalizt Dome. Seems' starting 11 was the same 11 he has selected ever since being made manager of the Jaguars after BoF71. Vice-captain Harold Gylfisson in goal with a back 4 of Aron McKenna, Mohammad Kallah, Nicolas Umkala and captain Jeff Donaldson. A midfield 3 of Pedro Vasquez, Peliniho and David Johnson was just behind "the Big 3" TJUN-ian attackers of Vladimir Podolov, Joe Green and Kepo Ulawaya. The 4-3-3 in its purest form. Valens, on the other hand, preferred the 4-3-1-2 with Ferret Tioux starting between the sticks. The back 4 was Kai Xiao, Henri de Aea, Giles Ken and Valence Ilya with a midfield 3 of Kaytlyn Victoriane, captain Enigma Armageddon and Jazmin Dragana. Hermaeus Mora played as an attacking midfielder for the front 2 of Keira Andisori and Olimpia Vidal. Both sides knew what type of game they wanted to play and, as such. many expected an entertaining fixture once it kicked off.
Once the game did kick-off, the National Stadium was fired up and ready to see this extra game of TJUN-ian will. Both sides knew an advantage in leg 2 was a must and so the game was tight from the get-go. In the 18th minute, someone did break the deadlock: Vasquez duping Kai Xiao to send the ball to Podolov, who made no mistake in the one-on-one with Tioux. The crowd erupted (the home crowd at least) and TJUN-ia were suddenly on top. A few minutes later though, the Anomalies would get an equaliser: Mora's free-kick couldn't be intercepted by Gylfisson. The rest of the half was tight, with both Green and Vidal missing chances, but half time arrived with the scores still level.
The 2nd half would be just as tight as the first, and both teams wanted the win. Even though they would have the away goals advantage if the game ended 1-1, the Anomalies continued to attack and the Jaguars did the same. Archer Andrews had an effort blocked by Tioux, Mora would have the same luck with Gyfisson. But someone had to break this deadlock, and it came down to Mora's replacement on the field. Mimi Kuzami countered a TJUN-ian attack brilliantly, leaving Victoriane to slot home with ease. It would be lead the Chromatiks would hold onto, despite the best efforts of Ulawaya in injury time.
A home defeat is sad for TJUN-ia, but it was expected by many considering recent history. Chromaitka will now host game 2 with not only a goal advantage, but the advantage of the away goals rule. For the Jaguars, the road to Newmainstan and Drawkland was falling away but it wasn't out of the picture at all. We didn't do a Squornshelous and lose 6-2 at home, so at least that's something. The aim is simple: TJUN-ia must win by scoring 2 goals or more in order to advance to the great tournament itself. Hard? Yes. Possible? You bet. GO JAGS!
PLAYOFF - vs Chromatika (36)
1st Leg (National Stadium, TJUN-ia City): L 1-2
2nd Leg (Capitalizt Dome, Chromia):
4th: WBC50, IBC31
Quarterfinals: IAC9, ARWC1, RLWC19, IBC30, ARWC2, WB41, IAC12
Ro16: BoF71, CoH77, WBC49, IACX, IAC11, DBC49, WCoH42, DBC50
Ro32: CoH78, CoH79
Other: Bowl Winners (AVBF6)
NSSCRA - Jaguar Racing: #07 Michael Stefan (2 Wins)/#64 Alfonso Mercado/#03 Maddison Riley-Jones (S10 T2 Champion/3 T2 Wins)
WGPC: #11 Lane Carter
NSTT: 3 Singles Titles (2 Runners-up)/1 Doubles Title (1 Runner-up)
UN - You all know what this is (U1)
TJUN (Ta-Jun) - An organ of the UN that focuses on "international role-play" (i.e. USA = Fang the Sniper) (U2)
TJUN-ia (Ta-Jun-ee-a) - The testing grounds of TJUN members, but operates as an independent nation. (U3)
Graintfjall
Diplomat
Posts: 658
Founded: Jun 30, 2020
Inoffensive Centrist Democracy
Excerpts from The Fußball Show with Jessika Jünkindietrünk.
Jessika is showing highlights from the first match, concluding with Alexander’s late save that denied Kate Huitema-Omeasoo.
“So Petter, a good save from Alexander Leonardsson there at the end. There’s been a lot of talk about the offense that will be needed to score in this match, but how important will the defense be?”
Caption: ‘Petter Erlingsson, Steinaux’
“Very important, Jessika. Remember that if the Snow Wolves lose, it won’t matter how many they’ve put up at the other end! And a 1 – 0 win, either way, would be enough. So Alexander and his defensive four need a strong game.”
“You say defensive four, and this is because we are suspecting Ketilbjörg Aunosdóttir to miss out as Dan Antoninsson will start, part of this attacking formation. Is Ketilbjörg a scapegoat, Olle?”
Caption: ‘Olle Tobiasson, journalist’
“Not at all. Her performances through the tournament haven’t been that strong. I suspect she will end up seeing the field at some point anyway. A lot is going to depend on the midfielders and they may run themselves ragged trying to keep up with quality players like Marco Hertel.”
“Petter?”
“No, I think Ketilbjörg should be reassured it’s this particular situation that’s sidelining her, not a reflection on her play necessarily. A more orthodox 4 – 4 – 2 will let Dan provide some support to Jason, who’s probably going to be key.”
“We started talking about the goalkeeper and we’ve made our way to the center forward.”
“Well, he’s by far Græntfjall’s leading scorer – if we need goals, I can think of no one who’s under more pressure to deliver.”
“…which is why Amanda Guttisdóttir has been such a colossal disappointment,” says Olle.
“And now, I’m delighted to say we are joined live from the training camp by Amanda Guttisdóttir!”
As Amanda’s face appears and Jessika moves over to talk to her on the high quality state of the art screen, Olle turns a shade of off-white and tries to hide behind his moustache.
“Amanda, thank you for joining us.”
(On a slight delay.) “Thank you Jessika.”
“What is the mood in the camp like?”
“Well, I’d be lying if I said there weren’t some nerves. But Mr. Brynjólfursson has spoken to us about the occasion and told us to see it as an opportunity. This is, frankly, not a position we considered ourselves likely to be in when we began this process – playing for a potential World Cup place!”
“I’m sure you know there will be many here wishing you well. Have you received any particularly special words of encouragement?”
“Yes, we had a special Doom call earlier from Ms. Michaelsdóttir, the foreign minister, and she had some very nice, encouraging words for us wishing us well. And of course we’ve spoken with family and loved ones. But now we’re really trying to seal off and mentally prepare for the match.”
“Of course I understand.” Jessika turns to her guests.
“Hello Amanda, Petter Erlingsson here. Good luck tonight. There’s been a lot of talk about having to play a more attacking style tonight, you could be key to that. How much pressure are you feeling?”
“Hi Petter, nice to hear from you and thank you. I think I feel a little bit more pressure after you’ve asked me that question! But no, I don’t think it will all be down to me. Certainly I know I have a big part to play, but we have to work together as a team. All of us have our assignments tonight and we’re confident we can adapt and show off a slightly different side.”
Jessika coughs.
“Hi … Amanda … Olle Tobiasson here. Good luck tonight. Um …”
Even projected from Quebec, the force of scorn in Amanda’s eye-roll is overpowering. Jessika quickly jumps back in.
“Amanda, what is the fan support like there?”
“Small but I am sure they will make themselves heard!”
“So there will be a few wolf howls tonight?”
“Let’s hope so.”
“I am sure there will be many more in Thing Park and Blue Square, where there are big screens being set up. And of course many smaller screens in homes, bars, workplaces…”
“Really helping me relax here ahead of the big game, guys…”
“Haha, sorry. Well, Amanda, thank you very much for joining us. Any last words for the fans?”
“I just want to say that this journey we’ve been on has been about more than goals and games and grass. It’s been a forging of our national identity. Two years ago, Græntfjall didn’t even have a football team; twenty years before that, Græntfjall didn’t have its independence. Guys like Vilbjörn and Ralf are old enough to remember that, to have come of age fearing they might never play under the blue-and-white, let alone in the blue-and-white. And so, win, lose, draw – whatever the result, the Snow Wolves will leave tonight very proud of all we’ve achieved.”
“Thank you Amanda.”
“But fuck I hope we win…”
Solo: IBC30, WCoH42
Co-host: CR36, BoF74
Champions: BoF73
Runners-up: DBC49
The White Winter Queendom of Græntfjall
GS SuperSports+
Independent Athletes from Quebec
Attaché
Posts: 99
Founded: Mar 20, 2020
Left-Leaning College State
Running Up That Hill
PART 5 (33) - Transatlanticism
‘Well, they certainly spent a great deal of time welcoming us,’ Eileen told Asher, after the military parade was finally finished. With a few hundred-thousand people having gathered for the occasion, they were really feeling squished and felt that this was a right time to get out and maybe chat for a bit more.
‘It’s flashy, with lots of soldats, tanks and fanfares.’ Asher chuckled, trying not to make too much of the comment. ‘At least we could clearly see those lines along the municipal squares.’ In all honesty, neither of them really paid much attention. Distracted and tired, they were just ready to visit the National Museum of the Sicoutian Resistance before heading back to the hotel for the night.
Neither Asher nor Eileen were the fans of so-called military marches. Sure, they had both went to the Royal Wedding Parade between Christine II and HRH Caspian, the Duke of Halifax, but those parades, of anything royal besides the funerals (and even they would often be accompanied with the galas and balls, as the late monarch and his late father had both desired), were jubilatory. Instead, whatever they had to go through for a couple of hours, with various Sicoutian chants narrating the greatness of the Sicoutian nation...it was far from what they had really desired.
‘So….apparently they asked me to be a ceremonial pitcher for tomorrow’s game.’ Eileen said, trying not to make too much out of it. ‘What does Mr. Catcher think of his beautiful, talented girlfriend pitching in Val-Maurice tomorrow?’
She knew Asher wasn’t going to be so comfortable with bringing sports when in unfamiliar situations, so it was all too clear to both of them that Eileen did this to catch his attention. For a guy who loved all forms of sports, and wrote about them to make half his living, Asher was particularly careful not to say too much about it. It was one of the things that he had long conditioned himself the first time they had dated, and accidentally kept around with him since then.
‘Oh, I have three words, Eileen.’ Asher gestured a pause. He was relieved at least- it’s not like the higher-ups had only expected her presence and not his, but then that’s just his selfish self speaking mostly. ‘You, Go, Girl!’ He smiled and gave her a loose hug, before quickly letting it go in the middle of the street. ‘That’s a fucking dope news...and they said tomorrow as well, right?’
‘Yeah.’ Eileen smiled graciously, her hair slowly strumming as if they were the piano keys made of ancient Ivory pieces. ‘Honestly, it’s an unexpected but genuine gesture that they asked over breakfast earlier today. Not usual we see a minister coming all the way to Edmundston, they said, so I just decided to...agree with it.’
‘That’s great….I’m glad nothing went badly with it.’ Asher took a breath in relief, after realising that nothing was lost when he failed to appear in time for breakfast. ‘Do you need any catcher, my dear Plongeon?’
‘Stop being silly, Ash,’ Eileen stopped midway, raising both her hands before moving them to her shoulder. ‘We’ll have time after dinner. Let’s just go on walks, while it’s still cool outside...as grandmaman used to say, Sicoutian summers are gonna start soon and those, you see, will be hot and blasting barnacles.’
Right….Asher recalled as he slowly gathered his thoughts. He still remembered how this new, Portnoian Sicoutimont, even with his preexisting knowledge and previous degree in history, was still a country he wasn’t so familiar with. It’s different than how it was for Eileen, who was familiar with its folklore and how some of the antique objects still remained in the De Ramaut Place, situated in the Highmore District of Montreal. He remembered how the place, though dusty and filled with enough relics from comedy of errors (only two of the eight rooms there were occupied when he last visited there with her last October), held that allure. Something timeless, foreign, yet long dusted, perhaps dating back to the last hurrahs of the Quebecois Commonwealth and many old regimes, before they were to face the test of time.
Probably because we can’t get it back, he thought to himself, as they started walking again to their hotel, to pick some stuff up before they were to head to a museum. While he was sure Eileen knew of the changes that had occurred over her life, with Sicoutimont being a victim of unexpected climate changes, a revolution and transformation into an authoritarian state, he wasn’t sure if those allures were still going to greet them, as they eventually head north to Val-Maurice and Lac-Drouin and the battlefields in coming days. Heck, even the climate may not even be the same...
...Perhaps what’s there at that dusty house may be the most of what her and I still grasp from old Sicoutimont….I don’t know.
Later that night...
‘Well, the day in St-Simeon’s quite something, no matter how you’d wanna put forth the propaganda side of things.’ Asher Lundrigan was resting in his hotel bath, after a long day in St-Simeon, on the Eastern coast of Sicoutimont. ‘Their insistence, for one, is something to say.’
He was having hard time trying to explain how the day went, from the Sicoutian military parades held in the middle of St-Simeon’s downtown or the afternoon tour of the Palisades, the famed iron castle built by the ancien regime in the 1800s. The confusion, of course, was partly over whether the locals seemed to have a particularly unilateral way of understanding their history, or whether the Quebecois guests were in disgust over how their ancestors’ service was left viewed on the sidelines.
Now, having decided to go back to his room earlier than intended, Asher was on phone with his college friend Charles, who’s been working on the House of Commons under the conservative (author’s note: this means Centrist in Contemporary political scale) Parti Quebecois. ‘Their understanding of history, factionialist at its very heart, is something that some may sympathise in, but it is not something that can be endorsed, I’m afraid.’
His friend wasn’t surprised. ‘Perfectly understandable. Their nation runs under a slightly different governmental model to start with, and we know this means that certain loopholes and telephone courts could exist.’ Charles, long familiar with the rhetoric and the public awareness of it, was in his parliamentary office with a glass of gin, trying not to speak too out of fashion. Having a long day at his office, he just decided to spend his overnight hours gaming and handling some documents for leisure. ‘You know, those are usually the most fun ones to talk to.’ Charles responded, aware that this talk was being recorded. ‘It’s like having a cookie filled with…..’
‘Similar, though I do not have the right term for you.’ responded Charles, trying not to spill his glass into the folder placed right above his table. ‘Explosives’, they were written. ‘You see, Sicoutian French is neither of our strong suit.’
‘Certainly not, because the vocabulaire there is entirely different from what Academie Quebecoise stated.’ Asher wondered how many days Charles spent in his apartments as of late. He remembered how Charles, during their entire undergraduate days, used to commute from a decent, 4-room apartment on the South Shore, which he and his brother inherited from their parents at the penultimate year of Lycee-de-Levis. He wondered if the apartment, which was as untidy as a large flat of two sociable undergraduates could be, was still under the same shape. ‘It’s almost as if the language there...had remained fossilised on its own. Which is quite fascinating, to say the least, if you are a linguist.’
‘I could imagine so.’ Charles responded as he turned his chair around, before starting to drink from his bottle of Gin again. ‘Anyhow, it must be fun getting to travel, and see how the time froze.’
‘I guess that may not be a bad way to describe,’ Asher responded. ‘The blocks of old apartments was one thing on the outskirts at least. Architectural marvels, those blocks- quite reminiscent of Prof. Salmond’s ‘Early Origins of Apartments and Urbanism’ course. You’ve taken it before right?’
‘Wish I did.’ Charles started drinking again. It was a quiet, perfect May night in Quebec City, and he knew this was the right time to do so. ‘Must have been that semester when I was too busy not to stare into suits and skirts-’
‘Oh you.’ He chuckled. ‘You need to stop drinking too much while gaming about at times. It’s never good for you, especially when you chat on message boards, be on phone and roleplay. Have you never learned from what happened last Halloween?’
‘Oh right- that time when I shared to a bunch of online strangers a picture of me in work uniform with few MPs? Yeah, that’s fine. Doxing’s not really my concern, you know. Parliament could care less if I game there, granted I polish the speeches and bills as asked.’ The security risk when travelling to Sicoutimont had been no secret to most high-profiled Quebecois visitors, who were always given specific governmental instructions and pamphlets to avoid certain outlets and websites when in the country. Both Asher and Eileen, due to different reasons (former being a journalist and latter actually being a high-profiled individual), were informed of this way in advance and did their best to stay out of line. So far, it’s been working. Of course, this did not prevent either from being not as careful with their phone calls. ‘People have tried to accuse me of shit over some past life grudges in those forums too. The classic St-Croix problem, where everybody knows each other, or everybody envies each other. So petty and silly- I get tired of them because it’s the same ole ‘no you can’t because year 2035’ bullshit- but that’s where I reserve my pity for.’
‘Easy. But then, pity is easy to buy these days…..I’d rather if you tackle them for chuckles, as if it’s a rugby scrimmage,’ Asher chuckled, seeing that his friend’s back to his usual self. Now it’s going to be somewhat fun to talk to, that’s what he thought. ‘And just bite until every bit of the flesh comes out and you could see their bones. They may scream but you won’t be hearing them scream. You can’t hear people scream online to start with, unless you stream shit.’
‘No, no, I’m not doing that.’ Charles shrugged, his body slowly moving around. ‘I mean, you could already see me where I work- Parliament has their own channel for that, and it’s not like I’m far from it either. Now tell me, how does their parliament work?’
‘I haven’t been there yet.’ Asher found this as the right timing to head. ‘That’ll happen in Lac-Drouin, though I don’t know if there’s much entertainment to be found in the congress where the Sicoutian Patriotic Workers' Front dominates in all fronts…’
‘Still, it should be interesting enough to see how that goes.’ Charles said, also feeling like he’s ready to drink once again. ‘Call me back when you’re there, eh? Or just wait until Valladares if you want to. I’m not going anywhere.’
‘Sounds good.’ Then they hang up the phone. Asher stared into the sight just outside of him. He was still having some trouble adjusting to the new environment, but at least there was a possible element to take away. They had made it through two days of fanfares, fireworks and casual conversations with little trouble. Now, with the minimum expectation being set, the rest of the journey was going to be easier. Most importantly, they only had to spend three more days there.
Squornshelan Remnant States
Envoy
Posts: 280
Founded: Jun 25, 2018
Left-wing Utopia
### WC86 Post-Qualifying
Black-and-Reds to Skip Cup of Harmony
At a joint press conference, SCFA president Ailen Koirala and team manager Anne-Sophie Groothuis made an announcement that caught the nation by complete surprise. "After due consideration," President Koirala explained, "the SCFA has elected to decline its invitation to Cup of Harmony 78." The decision will end a run of five straight cycles in which the Confederacy was invited to, and competed in the Cup of Harmony. Competed, though, is perhaps too generous a word to describe the last two editions, where the Black-and-Reds failed to advance from the second round despite being the top seed in their group. This, combined with the team's underwhelming fourth-place showing in qualifying, seems to be the primary motivation for the decision to stay home this cycle. Groothuis decried a culture of complacency and entitlement which she still views as the primary reason the Black-and-Reds have failed to make the finals and break into the top 30. "There is an expectation for this team, and in the footballing culture of this nation. Perhaps we won't qualify, but we will always warrant a Cup of Harmony invitation. I think it's important to make a statement here that this is not our goal, and it never should be." She did not rule out the possibility of participating in some manner of events during the year or so before IAC 11 rolls around, but unequivocally voiced the opinion that her team was not at all prepared to enter a tournament that aims to crown the "Best of the Rest."
In another shocking move, Groothuis has also proposed that her team should attend the second leg of the playoff tie between Terre Septentrionale and the Imperium, who advanced to that stage by virtue of their second-place finish in group 10. The Septentrionaux smashed the Imperials 6-2 in Vogsphere in the first leg, so this game seems largely a formality, but she suggested her players could do with the motivation, and the example. "The Imperium has passed us in the rankings, quite frankly, because they've been a much better, harder working, and more consistent team for some time now." Groothuis said that she would not compel any player to attend, but hoped they would choose to do so. "I think there's a lot we can learn from attending the match in Trzebiegoszcz about the way teams that truly have the drive and desire to make the cup play," she insisted after being bombarded with questions by skeptical journalists present. Whether she actually proposes to have her team study the match for their own instruction, or if this is more about making a statement of purpose, it could actually prove just as effective as refusing to allow them the opportunity to attend the Cup of Harmony. Many across the Confederacy have been saying for years that this team needed a culture change, and Groothuis seems determined to be the person to bring that change, no matter whose toes she has to step on in the process.
The Confederacy of Squornshelan Remnant States
True Democratic Successor to the nation of Squornshelous
Trigram: SRS
KPB Ranking: 15.10 (60th)
World Cup:
Did not Qualify: WC81, WC82, WC83, WC84, WC85, WC86, WC87
Cup of Harmony:
Quarterfinalist: CoH73, CoH75
Round of 16: CoH74
Round of 32: CoH79
First Round: CoH76, CoH77
IAC:
Third Place: IAC8
Quarterfinal: IAC10
Second Round: IAC6, IAC7
First Round: IAC9
Other:
BoF68 Quarterfinalist
Squornshelous
Attaché
Posts: 68
Founded: Antiquity
Corporate Police State
### Cloudy, with a chance of Coup d'Etat
Sweyn Petrov read over the communique sent to him from the Emperor's staff once again, as if the text might yield something different. It seems our assessment of your aptitude was premature. The toothless bastard was going try to pin the playoff disaster on him.
It had well and truly been a disaster. Six goals conceded at home! It had been eye catching enough that pundits the multiverse over were talking about it, and other teams on the wrong side of the aggregate were consoling themselves that at least they hadn't pulled a Squornshelous. That artifact Jaffacake had piled on as well. It would take immortality to have as much spare time as that man seems to come up with. On top of the humiliation of the lopsided score, Chaika had gotten himself suspended for the second leg, leaving Lagerkron to choose between Breiner and Cherenkov at left back. Personally, Petrov favored Cherenkov, but he saw little point in attempting to influence Lagerkron's decision. Needing five goals and a clean sheet to advance to the finals, or some equally specific and absurd scoreline, losing a major contributor from the backline was yet one more kick in the teeth. Who lined up for the Imperium would make little difference in this, a match that would soon be lost to the dustbin of history.
The communique wasn't really about football though, Petrov knew that much. A convenient excuse to block my advancement. He had, perhaps, underestimated the Emperor's information network, and analytical capabilities. While few ordinary citizens might realize it, Minister of Sport had always been a powerful and influential position, facilitating forming connections with diverse branches of the omnipresent imperial government. Since the return to international play, it had only grown more influential. The (hang the revisionists, tradition trumped spelling in Petrov's opinion) Pschychoes' rapid rise through the ranks had propelled him ever more into the public eye. Something he had plans to make good use of.
Very well, Lord Emperor. You may check my advancement for now, but if you think I haven't prepared for this, you're a fool. We shall see how comfortable that throne is in a few months time. He picked up the phone, and began to dial.
The Imperium of Squornshelous
Rightful Overlord of the Rogue Provinces of West Squornshelous
Trigram: SQU
KPB Ranking: 22.41 (32nd)
World Cup 31 Champions
Runners Up: WC15 & WC38
Third Place: WC20, WC25
Semifinalist: WC18, WC27
Quarterfinalist: WC5, WC11, WC12, CoH6, WC22, WC30, WC32, WC33, WC34, WC40, CoH77
Second Round: WC6, WC7, WC9, WC16, WC21, WC23, WC24, WC28, WC36, WC37, WC39
Group Stage: WC8, WC10, WC13, WC17, WC19, WC26, WC29, WC35, WC41, CoH76
Worst Day of My Life: WC14
Other days that were not the absolute worst, but when we also didn't qualify: WC84, WC85, WC86
South Covello
Envoy
Posts: 245
Founded: Nov 24, 2017
Left-Leaning College State
Beloved Goat Ruth Bleater Ginsburg Murdered by Anti-Goat Extremist
Beloved goat Ruth Bleater Ginsburg, who formerly belonged to Lord Almighty Gregory's menagerie of animals and Squirtles but later became a symbol of the anti-Gregist Revolution after she was seen draped in an Interim South Covellan Flag and the Robes of Justice and celebrating with Rebels as they raided the false deity's mansion, has been killed by an anti-goat extremist who claimed loyalty to an unspecified movement of goat eaters in Banija, South Covello Federal Police reported.
According to video footage, Ruth Bleater Ginsburg, who lived on an animal preserve with the other members of the former menagerie and their offspring, was frolicking in the fields eating grass, newspapers, and tin cans when a man streaked into the field and stabbed her with a knife, then attempted to make her into goat meat before being tackled and arrested by Park Police. He was taken to a field office and identified as Robert Guilloma, a South Covellan citizen. Upon interrogation, he proclaimed loyalty to Banija and stated "Goats are Food, Not Friends. I want to eat Ruth Bleater Ginsburg because humans are glorious and goats are our food." However, authorities say it is unclear if he actually has ties to Banija or if he is just pretending to have ties in order to intimidate goats and friends of goats. An official who spoke on condition of anonymity compared it to terrorists in the day of Reino Kulseth who would proclaim loyalty to the Christian State terror group. "Do they actually have ties or are they just trying to scare you? We're still investigating here, we'll let you know when we know more."
With World Cup 86 starting soon, Benitez also took the time to remind the Banija National Soccer team that a South Covellan witch doctor had previously placed a curse on them that they would never win the World Cup until the nation bans goat-eating after he was not allowed to enter the PG Arena with his pet goat for a World Cup match in Banija during World Cup 81. #TheCurseLives is now trending on Twii.tur, as are #RIPRBG and various other memorials to the beloved goat.
The Banijan Embassy was also unavailable for comment as of press time.
Thankfully, no other animals were physically harmed, although a Squirtle who watched the vicious attack is reportedly extremely traumatized and refusing to come out of his shell. A Squirtle Therapist is attempting to deal with him and make him happy and unscared again.
When asked how she would deal with the situation, President Luisa Benitez said Guilloma would be "dealt with harshly" and that if he actually did have ties to Banija, the South Covellan government would "certainly seek reparations." Meanwhile, Guilloma has been charged with one count of first degree animal abuse, one count of illegally causing death to an animal, one count of entering government land to commit an unlawful act, one count of murdering a protected animal, and one count of clogging a toilet in a national park since he apparently took a bathroom break before murdering Ruth Bleater Ginsburg and left an awful mess in there. He was released on his own recognizance, against the wishes of prosecutors, with Judge Martha Mays stating that criminals looking to make a point normally don't skip out on trial. She added that if the prosecution could prove Guilloma actually had ties to Banija and wasn't being blusterful, she would reconsider and potentially even remand him.
Stay tuned for more on this developing story.
Last edited by South Covello on Thu Sep 24, 2020 4:48 pm, edited 3 times in total.
Independent Athletes from Quebec
Attaché
Posts: 99
Founded: Mar 20, 2020
Left-Leaning College State
Running Up That Hill
PART 6 (34) - Just A Game
Mmmmmm......Asher gave a quiet buzz through his mouth.
There was something with Val-Maurice that dearly comforted him. Perhaps that’s why the Parc des Martyrs gave him a slight throwback when Asher entered the Press Box. Having been built in 1924 and routinely renovated, though not in the past twenty years, the stadium still holds up like an old relic from the town’s heydays as a mining town. The arena’s tall, green stands on the outfield, for one, always drew many eyes at how difficult and pitcher-friendly it was, and with the raucous cheer of the GVRT fans at home, you couldn’t help but to wonder how it would have been like on a different, less stressful era for everybody.
The former Sicoutian broadcasters, Fidel and Jacques, who were there with Asher and few others today, told him that Val-Maurice was the heart of Sicoutian resistance for that reason- the pride, the tenacity, and the urgency- and Asher couldn’t help but to agree to the matter. Perhaps that’s why the stadium still retained that pre-regime atmosphere, though it’s likelier they just didn’t give enough darn, Asher thought as he grabbed a glass of beer. The war and peace that coexisted in the city certainly had him think a bit to answer, and he was taking more time to appreciate the legacies and whatnot. The city, in some ways, was being more true to the past than any of the previous ones had been, and remained so, in the ways that classics students would talk about (when not drunk) in their classics field courses.
‘How are you finding your time in jolly, good Val-Maurice? Party officials have told me you’re the only Quebecois journalist allowed entry on the Cross-country train this year.’ The old, grey-haired Jacques asked in a mellowed, baritone voice of a 74-year old man. His voice, after all the years he had spent first on his podcast with longtime friend Fidel, still retained bits of the fame from the days of old. ‘I hope you’ll get to enjoy the match today- not often can etrangers say they had watched the match between le Metropolitain and good ole GVRT, the greatest baseball team in Sicoutian history.’
‘It’s indeed my honour to be here, thank you for inviting me today.’ Asher nodded, while watching a children’s choir sing the Quebecois, Cassadaiguan, Valladar and Sicoutian national anthems. Of course, the decision to snub the old anthems of the Novopetrogradian Soviet Union was noticeable, but what else were there to say about it? It was evident from the old times that the NSSR and all forms of Sicoutian ancien regime didn’t get along, and it wasn’t like that much had changed between the two when the Second Sicoutian Revolution happened when Asher was a toddler. While the culture of old may only remain in the pictures and archives, the systems that held the old world together continued to serve the new one.
Knowing that this was a formal occasion, with various dignitaries and sportsmen present, Asher stuck with a solid, pale suit with red and black tie, that’s topped with a 100th Commemoration Tour badge that he had received at the very first day of his travels on customs. On most occasions he’d have donned something simpler, whether it be a Kingston Tigers, Raptors or Quebecois national team jersey on summers, or a simple cardigan with sports jacket added on top, but that, of course, was on the Sicoutian Embassy’s List of What You Shouldn’t Bring.
‘How did you, and other Sicoutians, find all those years of isolation from the international sporting scene?’ Asher asked, as they watched the anthems end and Eileen slowly step onto the diamond for the ceremonial pitch. Unlike Asher, Eileen was gifted with the gold-and-black jersey of the GVRT Val-Maurice, the team of the Sicoutian Army reserves based on the metropolitan regions. ‘I remember reading when I was little, about the Sicoutimont of old, and how its sporting legacies used to be something to admire. A world cup qualification, finishing second in World Baseball Classic, et cetera-’
‘Hmmmm, it’s a tough one to answer, young man.’ Jacques responded, the long buzz seemed to suggest that it was a difficult question to frame around. ‘You see, what you tell me of the glorious days of Sicoutian sports….those were indeed the glory days.’ He responded, trying to find the right thread of thought, in order not to get himself in any more trouble with the authorities. Of course late Fidel, his good friend and the eternal broadcasting partner, would have given Asher a better answer but he had already passed away a year ago from stomach cancer.
‘Sometimes, you wish that those days would remain forever, but it’s not as easy as you’d think. All good things come and go, like how the elderly used to say when Fidel and I were younger.’ That was a good enough detour, Asher nodded, as he listened to the retiree speak. ‘It’s a good thing you asked, because Quebec was absent in international football for some time? Wasn’t it?’
‘Yes, though we did come back about three years ago.’ Asher nodded, trying to remind Jacques that the Grim Reapers, after many turns and whirlwinds around from a domestic scandal that led into the dismissal of the now-disgraced Prime Minister by late Jaccques IX. ‘It did take much effort, but at some point, the Grim Reapers had to come back, I guess.’
‘Good.’ Jacques nodded in satisfaction. ‘That’s sometimes how life works. Everything comes and goes, but when things do matter, and only if they really do matter, the people will want them to return. Sooner the later, the national teams will come back playing and the past unifies with the present. That’s the way to go.’
‘I’m delighted to hear.’ Asher understood where the old man was going with it, as they continued to watch the crowd cheer at Eileen, the visitor throwing the ceremonial pitch of the day. The return of Sicoutimont to international baseball, that’s been on everybody’s mind at the Concord Heights head office, when he had visited Concord Heights on a work trip a month ago. While there were no words on when or where the announcement was going to be made, or if it’s going to be made at all, the prospect of their neighbour and a past ally had gotten everybody to take a particular notice at any possible news coming from the other side of the border. Now Asher was understanding it, on the internal motive of why she was throwing a ceremonial pitch in middle of a city that particularly struggled, but still held the crown of Sicoutian baseball. It may be a struggling mining town, especially in the era of eco-communist society run by Alphonse Portnoy, but there were still enough in the city, whether in its battlefields or the diamonds, to be carried and continued from.
Of course, that wasn’t on the old Jacques’ mind, as he held a mild cup of Americano with him. ‘Anyhow, I am delighted he throws a mean curveball.’ Jacques said, giving a loose wink at Asher. ‘She really knows how to pitch! Not fast, but it's bloody filthy.’
'She does.' He smiled, remembering that it happened to be his girlfriend who threw the ceremonial pitch. No matter how he would have described it, he would have said the same as well.
Last edited by Independent Athletes from Quebec on Thu Sep 24, 2020 7:41 pm, edited 1 time in total.
Newmanistan
Negotiator
Posts: 5274
Founded: Feb 17, 2005
Compulsory Consumerist State
[align=center]WORLD CUP 86 - NEWMANISTAN/DRAWKLAND
Cutoff for Qualification Playoff Leg 2
Didn't look up a video. No one watches them anyway, right?
Just breathe in, breathe out, if you are one of the involved teams.
Last edited by Newmanistan on Thu Sep 24, 2020 7:36 pm, edited 1 time in total.
Six-time World Baseball Classic Champions
NEWMANISTAN SPORTING ACHIEVEMENTS:
CHAMPIONSHIPS: DBC 4; 27th BoF; CoH 34, 36, & 37; Oxen Cup 12; WBC 10, 12, 15, 17, 41, & 43; IBC 4, 5, & 29; CE 26; WLC 1
Runner Up: DBC 5 & 6; Oxen Cup 6; WBC 7,9 11, 14, & 45; IBC 1; WB 4, 6 & 34; WLC 2 & 3
World Cups qualified for: 46, 48 (R of 16), 49, 50, 54
Hosted: WORLD CUP 49, WB 1, 2, 5, & 35; WBC 8, 11, 14, 19, 38, 44, & 46; CoH 33, 35, & 39; CE 25, WLC 2, 4 & 5; WCoH 10, IBC 24, NSSCRA, Multiple NSCAA Basketball Tournaments, and a horse racing series
Murphtannia
Attaché
Posts: 84
Founded: Jul 06, 2020
Democratic Socialists
### A final win to end a good campgain.
Match day 18
Murphy pulls up his socks, the game will begin soon, he thinks to himself *Well we didn't qualify, but it's been a good campaign, hell I've become the all time leading goal scorer. I suppose the pressure is off tonight, neither side has anything to play for but they are so much better than us. Here comes the manager better start paying attention*
Murphy (the manager) speaks "Well lads this is it, the final game of our World Cup Journey. I know we didn't qualify but most people had us favorites to finish last, and we can still finish as high as 6th. So we have all to play for, I also have good news, it's been announced we've accepted our invitation to the cup of harmony, so go out there today and show everyone why they should be scared of us!"
Murphy (the player) watches as everyone gets up and walks out, they look like they could walk through walls at the moment. He is captain today but he's gotten used to the added pressure.
Murphy continues with the pre-match until it's time to kick off. Hew can hear the roar of the Savojorna crowd but he can also hear the Murphtannia fans, it's a shame we never won a home game he thinks to himself, but he can make up for it by winning here today.
[The 30th minute] Murphy runs into space, he shouts "long ball, long ball" to Cartlidge, soon he watched the ball zip throughy the air *you know the drill now, extend your foot, catch the ball and sprint like your life depends on it.* he thinks to himself, and he does get the ball and starts his break away. *Ok I have the ball, just got to keep going, wait where did he come from, oh ffs.* the Savojorna defender makes the slide tackle and takes the ball from him, the chance is gone.
[50th minute] *I need to be more involved, it's been even so far but I need to do something. Wait jenk is making a run, ok who has the ball, Pallett* Murphy turns, trying to make eye contact wit callum "Oi, I have a plan lob me the ball", sure enough the ball finds his way to him. *Ok I have the ball, but they have 3 defenders on me, ok ok I've done this before, hold the ball, keep holding, keep holding, there his legs are wide open makle the through pass* he makes the pass and jenkins easily gets on the end of it.
(cuts to commentator)
"It's the 50th minute, Murphy has just played a brilliant through ball into Jenkins, who is now 1 on 1 with the keeper, jenkins takes the shot GOALLLLLLLL!!!!!!, OH MY GOD JENKINS HAS JUST SMASHED IT INTO THE TOP CORNER OF THE NET, HE'S GOING WILD, MURPHY'S GOING WILD, WHAT AN ASSIST FROM HIM. MURPHTANNIA 1 SAVOJORNA 0, TAKE A BOW SON."
(cuts back to Murphy)
*Yes it worked!, I imagine that just made me quite popular, but now I have to concentrate, the game isn't over yet.*
[80th minute] *Ok I've been moved into a LWB position, the manager wants to hold onto this win, Oh crap he's got the ball past Owen, I'm going to have to try and get the ball off him* Murphy breaks into a sprint, desperately trying to get back but there is so much ground to cover, he can feel his heart racing in his chest, his lungs burn as he struggles to catch his breath, but he's so close *I can't let him get past, I'll have to go in for a slide, and yes! I got the ball* Murphy with the inch perfect slide tackle.
[Full time] *Yes WE DID IT! Thats a superb win for Murphtannia, I guess all there is left to do is make my mark on the Cup of Harmony.*
Last edited by Murphtannia on Fri Sep 25, 2020 6:03 am, edited 1 time in total.
Trolleborg
Chargé d'Affaires
Posts: 475
Founded: Oct 11, 2010
Inoffensive Centrist Democracy
Dear compatriots and our foreign friends! I begin this speech with the desire to share joy from the great success of sports in Trolleborg. Three years ago we started our path in world football, and during this time we have received a fair amount of bumps – in fact a lot of this. In fact, there are so many problems on the way and sometimes thing seemed so unpromising that several times the question of stopping to participate in sports on the world stage was raised. Dear fellow citizens, together we followed the few ups and many falls and failures of our sport, together we empathized with our players, supported them in difficult times, sometimes criticized them when required. Today we all have a big and well-deserved celebration – and damn every further doubts!
We would also like to share our joy with those abroad whom the play of our players and the passion of our fans, our inventions and innovations did not leave indifferent. Thanks to everyone who supported us and wished us luck. In a huge and harsh world, it is always important to feel that your work finds a response in the hearts of all kinds of people. Our success is yours too.
Our fate in the future may be diffucult or not, and move in every direction by thousands of paths, but I will express the general opinion of the inhabitants of Trolleborg: we hope that our success will serve as an example, as a guiding star, inspire many and many not to abandon a business that has just begun or is halfway through. Fate in the football world is not wholly blind, but it’s the sum of circumstances, many of which are strong and unfavorable to those who start their journey, but they can be overcomed. With a lot of work, perseverance and innovation, with addition of little bit of luck, it turns out that you can succeed even when the field was seriously tilted. Perhaps this is the rarest combination of circumstances that fell to us. But it is still possible nevertheless! Therefore, my advice to those who will go to from the very bottom to storm the mountains by our footpaths or by its own way: do not despair, come up with something new, which no one has done before you, endure failures in the first stages, and maybe you will be lucky too. The sweeter the victory, the more difficult it was to get it.
I wish to congratulate us all again, and let's hope that this success is not the last! Hail the victory!
Terre Septentrionale
Diplomat
Posts: 591
Founded: May 31, 2019
Ex-Nation
Trzebiegoszcz, Pomerania - Terre Septentrionale made it again! The team blanked Squornshelous 3-0 and advances to the World Cup for the second year in a row! The 2nd leg of the playoffs was played at Stadion Hydro-Polnocna (which means Northern Hydro Stadium in English). Every players and fans came back from Squornshelous without being detained.
Terre Septentrionale is the first team to score (obviously) at the 28th minute when Roksana Kolodziejczyk passed the ball to Sandrine Fontaine who passed backwards to Élisabeth Normandin. Normandin gave back the ball to Fontaine who shoots and she scores! The Nordiques increased their lead at the 36th minute when Roksana Kolodziejczyk dribbled with the ball and passed to Zvominir Kordic who kicks the ball and it's in! The score is 2-0 at the halftime.
Terre Septentrionale got a corner kick at the 75th minute, Yannick Ksiezniakiewicz kicks the ball in the penalty area and Wlodzimierz Waclawczyk heads it right inti the net. The final score was 3-0.
BOXSCORE
28th minute: Sandrine Fontaine assisted by Élisabeth Normandin
36th minute: Zvonimir Kordic assisted by Roksana Kolodziejczyk
75th minute: Wlodzimierz Waclawczyk (corner kick)
YELLOW CARDS
9th minute: Zdzislaw Brzezanczyk
48th minute: Anselmi Voronin
61st minute: Élisabeth Normandin
RED CARDS
None
World Cup 86
Terre Septentrionale is now officially qualified for the World Cup 86 held in Drawkland and Newmanistan. The draw will be held tomorrow evening. 18 group winners, 6 second placed teams, 6 playoffs winners (also 2nd placed teams) and 2 hosts are participating in this 86th World Cup.
Wlodzimierz Pietruszewski,
Senior writer,
La tribune Septentrionale
Nation name: République de Terre Septentrionale | Trigramme: RTS | Capital: Ville Jacques-Cartier | Maps
Ranks: Hockey: 20th | American Football: 7th | Baseball: 17th | Association Football: 23rd | Rugby Union: 21st
Champions:
Runner Up: Cup of Harmony 76, International Baseball Slam XI
3rd Place: World Volleyball Expo X, International Baseball Slam XII, World Lacrosse Championship XXXV
4th Place: Arena Bowl VI
World Cup participations: WC 85 (3rd place in group), WC 86 (3rd place in group)
TJUN-ia
Diplomat
Posts: 822
Founded: Oct 04, 2019
Civil Rights Lovefest
### A Message Left In The Capitalizt Dome Changing Rooms
Dear Chromatika,
Congratulations on your advancement to the finals of World Cup 86! We certainly played up to your level during the final game, but your talents on the field were just too much for us to handle - as it should be.
For TJUN-ia, it may come as a disappointment to come as close as you can get to the promised land only to fall at the final hurdle but, in many people honest eyes, they wouldn't have expected us in that position in the first place. After 6th in our debut WC and the disaster of WC85, no one knew what TJUN-ia was going to turn up here - and what we would do.
But we believe we have proven a point across these 20 games: that the Jaguars of TJUN-ia aren't just "another team in WCQ". We played some of our best football during this process and played some epic matches too - including our final 3 games against you and Tikariot. In TJUN-ian terms, this tournament was a great success and one to build on in the years to come.
So we shall leave Chomia as the same team who arrived here: a nation from The North Pacific looking to take its place among the elite of this sport. If we ever play again in the future, mark our words: The era of the Jaguar has only just begun.
Hope you cheer for us from afar in the upcoming Cup of Harmony! Go Jags!
Thanks,
David Seems
Li Xiu Ying
And the players from TJUN-ia
p.s. the URL links to a song from U1, sung by the Representative to Tajikistan in U2. It's TJUN in a nutshell.
PLAYOFF - vs Chromatika (36)
1st Leg (National Stadium, TJUN-ia City): L 1-2
2nd Leg (Capitalizt Dome, Chromia): L 3-4 (Chromatika advance 6-4 on aggregate)
Last edited by TJUN-ia on Sat Sep 26, 2020 8:36 am, edited 2 times in total.
4th: WBC50, IBC31
Quarterfinals: IAC9, ARWC1, RLWC19, IBC30, ARWC2, WB41, IAC12
Ro16: BoF71, CoH77, WBC49, IACX, IAC11, DBC49, WCoH42, DBC50
Ro32: CoH78, CoH79
Other: Bowl Winners (AVBF6)
NSSCRA - Jaguar Racing: #07 Michael Stefan (2 Wins)/#64 Alfonso Mercado/#03 Maddison Riley-Jones (S10 T2 Champion/3 T2 Wins)
WGPC: #11 Lane Carter
NSTT: 3 Singles Titles (2 Runners-up)/1 Doubles Title (1 Runner-up)
UN - You all know what this is (U1)
TJUN (Ta-Jun) - An organ of the UN that focuses on "international role-play" (i.e. USA = Fang the Sniper) (U2)
TJUN-ia (Ta-Jun-ee-a) - The testing grounds of TJUN members, but operates as an independent nation. (U3)
Kelssek
Minister
Posts: 2066
Founded: Mar 19, 2004
Civil Rights Lovefest
THE FOOTBALL ZINE ALL THE COOL KIDS ARE READING
AGREED PERSONAL TERMS SINCE 162 F.E.
Kelssek’s World Cup prospects: [internal screaming]
There’s never really such a thing as an easy draw at the World Cup, but Kelssek’s task will be an especially tough one as they’ll have to find a way past Farfadillis and Eura in the group stage to stand a chance at being mercilessly crushed in the second round. Also in the mix is a team of “independent athletes” from The Royal Kingdom of Quebec, who are not sanctioned as a national team by that country’s sporting authorities despite no other team existing, or consist of the remnants of some mass doping scheme, or something like that. Obligatory anti-imperialist slur here, royalist pig-dogs.
And there’s the dilemma for a third seed: you’re also done for if you don’t take the three points against the ostensibly weaker team – knowing that the putative top two won’t let up either. Taeshan and Equestria found that out last World Cup when Kelssek was the fourth-seed banana skin which also ended up slipping through.
The current generation have never won against any of these three teams, whether their athletes were independent or not. That’s to say, the only country Kelssek has a history against is Farfadillis, who beat us twice in the World Cup 81 qualifiers. Many of the team know Eura pretty well, however; Ruslan Demetriev, Damien Halliger, Cory Greenwood, Matts Bogdanor, and Loïc Maçon-Petrault are all playing for Euran clubs.
With that said, what kind of squad selections should Andrea Crowe make? Should picking players who can do a job against the teams you know you’ll be playing be a factor? We look at some of those fringe players and the case for and against putting them on the plane.
Phillip Lozic
Debuting for the national team this cycle, the Burnaby SC central defender is probably a lock for the squad given Kelssek’s lack of depth in this position. He showed he can cope against top teams when given 60 minutes in the recent friendly against Siovanija and Teusland. Picked up 9 caps during the qualifiers and did fairly well against Siovanija and Teusland.
James McDonald
The Rolalas FC player is certainly world-class but he plays in the two positions Kelssek has no shortage of talent in – centre and left midfield. As proven as he is in top-ten league, it’s hard to see a case for picking him over Colm Ó Tuathail, Patrice Champetier, Rémy Dionne, and Gabriel Lapierre, who all occupy a similar place on the pitch and/or bring similar skill sets to the table.
Loïc Maçon-Petrault
More likely to make the team than not, the question isn't really about whether he's good enough in terms of skill but whether he can deliver in the high-pressure moments. The youngster is showing some flashes of brilliance and could well earn a few minutes depending on the situation. But whether or not he does, this clearly isn’t going to be his last World Cup. A year ago he was a raw talent who really needed to get some seasoning at a high level, having spent some time in the Euraleague, he's at least on the way there.
Andrey Yusupov
The Strathcona winger’s emergence couldn’t have come at a better time for a national team that has not had a proper right-sided attacker since Blake McDonagh decided to retire from international football last cycle. While Patrice Champetier, Ashley Douglass, and Shaheen Taleb have all taken shifts in the position and are probably better all-round players, Yusupov’s advantage is that getting forward swinging in crosses on the run is his bread and butter.
Cale Shaughlin
CS Saint-Rémy’s number one is not the only possible third goalkeeper, but he’s the candidate who played the most in the qualifying campaign, so he’s probably in. May even get a chance if things go badly wrong since Cathal Gallagher hasn’t been impressive this year, and the way things have gone at Kirkenes FC, Kai Poirier may need to adapt to having a slightly less calamity-prone defence in front of him.
Nacuémiró Buené
The 19-year old Farfadillis-born Burnaby SC left-back played for Kelssek in the Olympics. He earned his first senior cap in the Siovanija and Teusland friendly match after breaking onto the scene during the league season. But this would be a really long shot, and we’ll have to say he won’t be in the squad, despite the fun narratives it would create for us journalists.
SIOVANIJA AND TEUSLAND 3 – Ribbeck (20’, 49’), Bahr (83’)
Heuser, Steimle, Jager, Niedermayer, Vesele (Lotbiniere 45’); Kramer (Bahr 45’), Mihailov, Bliznakov (V. Dimitrov 64’), Kogler (Kokalj 69’), Tauscher, Ribbeck (Ilev 64’)
KELSSEK 3 – Maçon-Petrault (17’), Custworth (26’), Dionne (60’)
Gallagher, Liadon (Bowler 45’), Curtin, Lozic, Douglass, Greenwood (Bogdanor 69’), Blueman (Lapierre 45’), Dionne, Yusupov, Custworth (Buené 45’), Macon-Petrault (Taleb 64’)
Match report: A warm-up match for both sides, giving several players a chance to show what they’ve got with national team managers due to decide their final squads for the World Cup. Phillip Lozic and Loïc Maçon-Petrault probably passed their respective auditions, and Nacuémiró Buené didn’t stand out in his 45-minute international debut but that’s probably a good sign all things considered. Defence needs work, but we already knew that, and the home side were noticeably better in the latter 20 minutes of the match.
KELSSEK 3 – Taleb (pen. 29’), Bowler (62’), Ó Tuathail (66’)
Poirier, Bowler, Halliger, Lozic, Liadon, Demetriev, Lapierre (Dionne 70’), Ó Tuathail, Yusupov, Custworth, Taleb (Lister 84’)
Mriin
Chargé d'Affaires
Posts: 442
Founded: Nov 17, 2016
New York Times Democracy
And now for the latest installment of Defecating Daffodil's series (youtube link) The Idiot Minute! The video you see here is merely a loose recreation of the actual video for perusers of the RL States platform; ICly Daffodil has much greater video editing skills than the author of this roleplay.
Welcome to the Idiot Minute, a short-form video series where I, Defecating Daffodil, show you how useless a discipline statistics is.
Wondering who’s the best team in the world right now? “Why that’s easy,” you might say. Banija are at the top of the KPB rankings! They’re the team to beat. But I have a much more interesting question: Who’s the best against the best? That’s why I’ve taken cues from Jeremey’s work in the spreadsheet mines and adapted some World Cup data made public by a brilliant researcher in Poafmersia. What I’ve devised is a “Strength of Schedule” calculation, popular in college gridiron and utilized by NSCF.
Well, I lied there--I actually made two formulas. One defines “strength” as KPB rank post-qualifiers, and the other uses points accrued during qualifying. On paper strength versus actual results, essentially. I’ve run the numbers for all 30 teams that qualified for finals, and maybe we’ll be able to make some predictions. We’ll first look at the results when based on rank. In this calculation beating a side gives you points equal to three times their KPB value, a draw gives each side the other’s KPB value, and a loss is naturally worth nothing.
Here we compare the traditional measurement of strength--world ranking--with their Strength of Schedule. Pasarga tops the list, which makes plenty of sense looking at their results--a second seed who topped their group and did the double over Eura, group 3’s top seed. Obviously in this calculation beating top seeds is worth major points, as that’s taking the biggest base KPB you can get and multiplying it by a whopping three. You might have noticed an outsider name snuck onto the list--Mercedini is third for KPB Strength of Schedule despite not qualifying. Beating Baker Park twice will do that. Nephara and Banija are two names expected to be up here from the sheer domination of their groups, with surprise package Trolleborg rounding out the top 5.
At the less-than-sharp end, we find Terre Septentrionale with a stunningly measly total of two-hundred and fifty-nine; fully thirty-eight points below the next lowest team, a resurgent but still struggling Chromatika. Kandorith is the third of three teams to not breach 300 on the scale, while relatively underachieving pot one sides Zwangzug and Kita-Hinode are fourth- and fifth-bottom. Seems they share more than a color scheme now.
Then we move onto the points-based Strength of Schedule, which works similarly but instead of multiplying by their KPBs you use their point total from qualifying. And now it looks a bit more like how you’d expect it. The complete and utter shitstompers take the cake here, with world number one Banija complimenting their absurd goal differential with an eighty point lead on second place. Nephara, Valanora, and the Holy Empire are all names you’d expect to be here--plus Pasarga popping their head up again. Do you give them the edge for being first-and-fifth, or the Nepharans for a steady second in both formulations? I’ll leave that for you to decide.
At the bottom, it’s much the same--Terre Septentrionale is the rump once more, with Kandorith and Kita-Hinode also returning. Equestria’s another underperforming top seed, and… Farfadillis is fifth-last?! They were doing just fine--ah, but they were in the clusterfuck that was Group 14. With four teams who were alive for the playoff spot on the final matchday, nobody accrued a particularly high point total for the Farves to profit off of. The weakest group of the bunch, at least going by point totals.
Speaking of groups, the group draw for the finals was revealed recently… and we can use our statistics to make up some bullshit about it! This diagram--with a handy reminder of who’s in what group--can be used to determine what the oh-so-often discussed “group of death” is. At the top we see a tie in average world ranking between the general consensus--the so-dubbed “AOCAF Semifinal” Group B--and Group F, who’s nations are perhaps a bit unfancied in the current consciousness but without a doubt carry a lot of class with them. Group H is also just a fraction behind, featuring Nationstates Trivia Challenge Champions Kandorith. At the bottom by a wide distance is Group A, thanks to Newmanistan and The Sarian who collectively have nearly fifty cycles since their last qualifications.
Now we’ll sort it by the average KPB Strength of Schedule. For the groups containing the hosts, they’re of course excluded from the calculation due to not playing any qualifying matches. And--good fucking lord look at that gap! Group G is by far and away the leader in average strength of schedule, with each and every team being in the top ten. And, spoiler alert: they lead in the other formulation, too. So you’ve heard it here first, folks: Nephara, Cassadaigua, Trolleborg, and Siovanija & Teusland are in for the battle of their lives in World Cup 86’s true Group of Death.
Nephara wrote:It's retro as all fuck, and it works like a charm.
Trolleborg
Chargé d'Affaires
Posts: 475
Founded: Oct 11, 2010
Inoffensive Centrist Democracy
Greetings, compatriots! It’s TTV from the Trolleborg International Airport. We will update your information with the latest news regarding our national team.
As you know, our team is one of the weakest in the tournament in terms of rating. She got a place in the fourth pot, and according to the results of draw receive assignment to Group G, the company got extremely difficult - Nephara (2nd place in KPB tables), Cassadaigua (13th), Siovania and Teusland (20th), and this, of course, clearly the death group – for us. We will play, by order of, with Cassadaigua, Nephara, Siovania and Teusland, and it will be in Drawkland, but it is not yet known exactly where. Of all three, we only dealt with S&T, and each time awry, be it the level of national teams or youth ones.
As our national leader put it, "under such conditions, any result other than three defeats will be considered a huge team success."
Our footballers left for the tournament yesterday. Despite the dubious prospects, the fans saw off their idols as if they had returned with a victory. First, the players of the national team, accompanied by the leaders of the country, meet with the people near the walls of the citadel in Trolleborg City
then they went to the airport through the crowds who had come to wish the team good luck, a huge crowd of fans of all ages was waiting for them on the spot,
so we had to postpone the flight for an impromptu autograph session that started right in the parking lot.
Then our team proceeded to the aircraft and went to their destination on a charter flight of “Trolleborg Wings”. A huge crowd of fans lined up near the airport with plastic sheets of different colors so that one from taking off plane could see the huge waving state flag, complemented by the inscription "We are proud!"
In the next day or two, a lot of fans who decided to support our guys in their difficult test, will leave by the same route. All in all, from 7 to 10 thousand people are expected at each match of our team.
Official information from the FA regarding the squad:
TROLLEBORG NATIONAL TEAM
Nickname: Trolls
Total official games balance: +56=19–24 221 goals for and 158 against.
In the World Cup Qualifications: +34=15–13 144 goals for and 101 against.
NATIONAL TEAM
Team has three kits, usually using green one
COACH
The team is still coached by Morten TROGHIL – triumphator of the BoF.
In this tournament, however, he will have to completely change the tactics of the team, since the attacking fast game we are used to is most likely not suitable for the circumstances of our debut in the World Cup. At one time recently, the coach experimented with defensive line-up options, but not very effectively. It will be interesting to see how well everything comes out this time.
Goalkeepers
1. Tomas Kot Kotsson (28). Flexible and agile, quick-thinking, he, alas, don’t make miracles in the international games, but no one in our league look much better between the posts. And as a main goalkeeper in the BoF, three qualifying campaigns including successful one, two IAC, Melayu and South Pacific Championships, he amassed precious experience that no one else can count on.
12. Klemens Gould (26). Second-choice goalkeeper, with some up and downs on his road.
22. Marten Windass (35). Experienced goalkeeper with a lot of talent. In the time, he was like a Kot Kotsson, and albeit now he wasn’t so fresh and fast as in his heyday, but he knows many secrets of goalkeeper trade. Probably, this is his last major tournament, and it’s already known that he will have a farewell minutes on the field somewhere at the tournament.
Defenders
2. Nikolaas Laste (29). He was reliable left back, who has a tendency going forward, but in the recent years prefer to concentrate on his defensive duties, with a great success in the WCQ.
3. Kasper Wolters (26). One of the team brightest stars, he is right back by default, but he is very strong, very fast, have a very good shot and even better crossing skill.
4. Torgeir Dushegub (29). Known as a Column, this tall and strong player usually dominates a box both in the air and at the grass. Being always calm and confidence, he projected aura of such qualities around.
5. Aien Troger (26). He is fast, strongly build and extremely good in tackling. It’s very challenging to move into the box past him.
13. Torben Bagger (26). He was distinguished by composure and the right choice of position in the games of domestic championship, and he was called up to the national team due to the breakdown of the two key defence players, proved to be solid choice, and recently he managed shouldered some established players out of lineup.
14, Heiko Aldekamp (26). Useful addition to the team, can play both right and left defender, preferably first one, moving fast and tackling hard. Probably he will have much more time than usual on the tournament due to the less-than-brilliant conditions of Nikolaas Laste.
15. Morten Levenkrans (31). Full left back in the beginning of his career, he than move to the center, having considered almost impassable in both positions. Albeit right now not the fastest man on the field, he is very good man-marker, having excellent header and can be very useful when attacks began. He make six goal assistances in the WC85 campaign, receive good press after CoH, one of the few players of the team to have one, and was one of the better players of the whole squad in the WCQ86.
21. Joran Du Riis(24). Recent addition to the squad, fits perfectly into it, showing himself as a cold-blooded fighter, although some lack of experience in international games visibly hinders him.
Midfielders
7. Rolf Valgren (29). Quick-thinking, with a perfect understanding of game, seeing everyone and everything in the field, and with brilliant passing skills, this man can bring ball everywhere he wants. It is very possible, that he will be one of the greatest midfielders in the history of Trolleborg, have he not lacking some technique and speed. He definitely add something from international experience, but it will be seen, how much.
6. Hendrick Groningen (28). The defensive midfielder, probably the best in his generation, he was not selected to play in BoF and joined team by way, proved himself worthy of a place in the team, caused a lot of problems to the attacking players of the opposing teams. His talent of intercepting ball in the middle and using every possibility to begin attacks was fully appreciated by everyone – apart, probably, of his opponents.
8. Finn Finnsnes (29). Beginning his career as a right midfield\right winger, now he play more like attacking midfielder, moving by the whole width of the field. He was one of the creators of the victory in the BoF and success of WCQ-86. Despite age, he is fast, swift, smart, good dribbler and excelled in set-pieces.
9. Morten Troll (25). He is a brightest star amongst midfielders in the whole country. Genius of game, with a wonderful dribbling, beautiful technique, excellent pass and extremely good shot, he won many games for his local team single-handedly, but also a great team player. Highly probably, he will be a player of generation, and very probably, best player Trolleborg ever had. He brilliantly justified all the positive reviews of his game on the BoF, helping a team many times in difficult situations with its smart actions, and continue to do so in WCQ84-86, with many решающие голы забив сам.
17. Ruud Kolding (28). Left midfielder, covering lot of ground with his unbeliveable stamina, can press forwards or go back as team need. Albeit seen as a substitute, he won the hearts of fans and a strong place in the first team with his perfect game in the BoF matches, never giving fans a reason to be upset by the quality of his game.
16. Brutil Mogrin (28). Can play either in the center or as a right midfielder, with a nerves of steel and cold mind he will continue be a useful backup for any of the midfielders if need arise. Usually he was used as a sub for Groningen.
20. Mortimer Torquelston (25). Another man who managed to get into the team, although experts had questions about his game in friendly matches. Skillful winger, who can play at either right or left flank, very mobile and prove himself as a master in change direction of attacks. But he probably will mostly sit at the bank at the WCQ.
23. Alexander Bornitse Zach “Zhita” (27). Originated from the far-flung north part of the country, not well-known for a excellent players, he was an idol for his compatriots and find a place in the hearts of other fans. Almost devoid of what was known as classical football tecnique, have rather unusual style of play and often creates goal opportunities from nowhere. One example was BoF, when his visibly insane pass decided the outcome of the game in the quarterfinal. But he played quite pale in the WCQ and was dropped from the squad thereafter, but his workrate in the last two local championships give him a chance to return. Until the last day it was not known if coach take him into the Drawkland, with some pundits pointed onto the slightly younger Kenneth Pathmoor, who put decent performance in the last WCQ games as a better alternative.
Forwards
10. Stuart Kerr (29). Quick, brave, skillful, and having ability shot from everywhere in the box right despite defenders best efforts. Brilliantly confirmed his reputation as a prolific scorer, winning a prize for the topgoalscorer of BoF and add to his tally not a few goals in the other tournaments. After injury year ago he looked like fully regain his position and reputation, altought everything in the international football is fluid and unsecure.
11. Torfinn Skaloed (27). Intelligent forward as it was, he can move very quickly, constantly ahead of the defenders, successfully escapes from markers, and in case of need he often going back and successfully acquire ball here to begin attacks. But in the international game his performance is not very consistent.
18. Tom Gulbranden (28). Build like a very tall stone wall, he easily can tramp defenders if he wish to do so, but his strong leg shot, without any preparation at all, give him other opportunity to have a points of popularity with fans. It is looked impossible, that a man of such a stature can bring himself in the air at all, but nevertheless he is – strongly jumping and struck the ball by the head with deadly precision. This is what he prove in the BoF games four time in nine games, but WCQ85 campaign was much less successfull for him, IAC games give us even less joy seeing at his actions, and only at Pan-Pacific Cup did he regularly featured in team squad and find a net often. WCQ85 and WCQ86 seen him recovered from misfortunes and his goals is speaking by themselves.
19. Karsten Severin (24). This fast forward, who can conjured goal from nothing, are quite impressive in the last four seasons, and he has not scored less than 12 goals in any of them, but it is not yet clear how successfully he will compete with already proven players.
Team in the brand new red uniform
TEAM GAME
Altought fast and spectacular attacking play is what team accustomed for, but situation here is very different. Having three much-more-experienced and heavyweight sides against it, Trolleborg team, obviously, will concentrate on the defence.
Style –5.
My opponent, if they RP first, may do the following:
Choose my goalscorers: Yes
Godmod scoring events: No
RP injuries to my players: Yes, but I estimate injury severeness
Godmod injuries to my players: No
Hand out yellow cards to my players: Yes
Hand out red cards to my players: Yes only for a last ditch fouls or two yellows, any other variants must be discussed with me
Godmod other events: No, but if you have something interesting in mind, let me know and we try to find a middle ground.
National psyche tell players to put everything even if odds are heavily against, so almost every match in the league played with high determination, and national team shirt obliged for even more. Unsportsmanlike behavior is highly discouraged even if in the interests of the cause, and national team players are required to meet the highest standards possible, so it is completely impossible to imagine a player of the Trolleborg national team faking injury or begging for a penalty. But players are not easily bullied, though, and more than able to stand up for themselves both as individuals and as a team, and have a strong inclination to go tit-for-tat in case of excessive brutality on the opponent’s part.
OUR FANS
Fans is a pride of Trolleborg and almost part of the team. Having already earning themselves a name and a good reputation at every place where national team playing last year be it basketball, hockey, rugby or football, they surely will make their presence seen and felt everywhere games will be held, with their usual pastimes like marching with songs, waving flags, try local cuisines, putting cultural performances and trying involve everyone around in all of this. They will give out some Trolleborg memorabilia like a ribbons, buttons, scarves and so on. As already become tradition, we ask host countries to allow a parade of fans with a live music before each game and a places for series of exhibitions devoted to gastronomy, culture and national traditions of our country. Also we ask allocate us some space in the hosting cities to make an exhibition of our cultural heritage and traditions with some traditional sport contests for a local citizens.
Trolleborg TV try to do it best to provide everyone with a 3D broadcasting of every game, altought you must understood limitations of technology, which was in their first steps. Problems with color shades, difficulties with communication channels and so on are rather often happens.
Last edited by Trolleborg on Mon Sep 28, 2020 8:01 am, edited 1 time in total.
Senator
Posts: 4351
Founded: Sep 19, 2008
The time is now,
By Chelsea Dufresne, Concord Heights Times
Here we are, on the cusp of the greatest sporting tournament in the multiverse.
World Cup 86 is about to begin, with the field of 182 ambitious nations being sliced down to the field of 32. There were many stories along the way, which have been told by beat writers from all over the World. A few nations that we are used to seeing in the World Cup will not be here. Others, who have been climbing up the charts, have made it here and are excited to make an impression. One good example of that is Cassadaigua’s first opponent once things get underway, as we meet Trolleborg. The competition is being hosted by Newmanistan and Drawkland, two nations that carry important symbolism in Cassadaigua’s soccer history. We know those two from different sports as well, but soccer is meaningful.
For Cassadaigua, the quarterfinal run in World Cup 49, co-hosted by Newmanistan, was big for our ranking boost, taking us from a nation that debuted in World Cup 43, and propelling us to not only being a nation that could qualify (which we did for the first time in World Cup 45), but could also have a shot at winning the whole thing. Little did we or the world know then, that we would end up winning the very next two World Cups, 50 & 51. Many of the faces on the World Cup 49 team did end up playing on the title runs, so the experience that they gained definitely helped out. Fast forward a bit, since Cassadaigua eventually took a long hiatus from international sports. When we returned, it was World Cup 78, and that was your typical unranked campaign, but in World Cup 79, as a pot six team, Cassadaigua quickly rose to the forefront and qualified for that World Cup, which was co-hosted by Drawkland. The two nations hosting the World Cup 86, as you can see, have played an important role for us, both in our earlier run and now in the run that we are hoping to make.
World Cup 86 qualifying went well, but so did qualifying in World Cup 85. In fact, since we made the semifinals in World Cup 83, we have been on a decline, dropping to the #13 ranked nation in the World. It’s a fair spot for us to be listed at, considering that on home turf we lost in the Round of 16 for World Cup 84 before the debacle in World Cup 85 and not making it out of the group stage. That was hard to swallow, especially for a team that scored 102 goals during qualifying and simply seemed to be on a high note. The team came crashing down to reality in the opening game of World Cup 85, thrashed 4-1 by the hosts, Mriin. Now, it’s World Cup 86, and this team that won their final thirteen games of qualifying is riding high. As Banija and Nephara have a bigger point total, that is a line to think about 13-0-0 to close out qualifying, with 50 goals scored and 19 conceded over that time. We could almost say that we called it at the Concord Heights Times, saying that a couple of the changes to the roster could mean the team doesn’t play to their full potential right out of the starting gate, but even we did not predict all that.
Personally, I try not to be opinionated too much here, and especially not in a competition like this, so I will defer to the comment of former superstar forward Hannah Ranucci, #33, who retired after World Cup 84. “I feel sorry for Trolleborg,” Ranucci began, “I love their fans and the amount of passion that they have for their team, but don’t think for a second that the team will not be shown videos and tapes of the 4-1 drubbing that Mriin gave our Fillies on matchday one of the World Cup proper last time. That feeling sucked out the life of this team, and is something that they have been thinking about every day since they left Mriin, eliminated in the group stage. The pain. The agony of that. Just eating at you, over and over again. Things seemed so promising, and they were ripped away from us in a heartbeat by Mriin. Now, the time has come. The time has come to redeem yourselves. Redemption for you as a player, and for those who were a part of the team then, but aren’t now. I would not want to be on the other side of the field for this game. Whoever that was, was going to deal with a Cassadaigua team that wants redemption greatly. Trolleborg’s number came up first. By that coming up first, as the pot four team, this team further knows that they must get three points. They can’t be in a situation where they must get something against Nephara on matchday three.” Ranucci, the captain of this team for several World Cups, leading up to 84 knows what she is talking about. Furthermore, keep in mind that players who are back now and were also on the 84 and 85 teams are also looking to overcome the disappointment of losing in the round of sixteen at Dagan Airways Stadium to Eura.
Trolleborg, our first opponent, is playing in their first World Cup. I caught a glimpse of a TTV (Trolleborg TV) broadcast, showing all of the people that had lined up to wish their players the best as they would make the long trip out to Drawkland. It is a special moment, as after all, it was just as special when we qualified in World Cup 45, and then again in 79 when we returned from the extended hiatus. We are at the stage now in our progression where qualifying is an expectation, so you are not going to see that amount of fans wishing us well. There will be fans that do so, but they are expecting us to do well, not just wishing it. From there, you can see why the Trolls can savor the moment and make the most of their trip to Sonnel.
Now, let’s talk about Sonnel for a moment. Going to Drawkland is something that we have done many times before. It is not always in soccer, but due to playing in so many different sports, as both nations do, we have a lot of experience in making this interstellar trip. Every nation will have their advisors, doctors or other smart people that will do what they can to assist the team, but nothing beats having people who have done it and can share what they learned. The Fillies have that, as manager Stephanie Sweeney has done it personally while coaching. Once on Sonnel, and in Drawkland, there is nothing intimidating or special about being on another planet, as many of these people have done it before, or closely know those who have. Trolleborg does not have the same experience and cannot fall back on this advice as well as we can. Does this mean anything? We’ll have to wait and see, but you can put it in the “food for thought” section.
After Trolleborg, the Fillies play Siovanija & Teusland, who will be coming off a game against Nephara. We’ll cover that game in detail when it is time to do so, but that is a very pivotal match. Trolleborg, after us, gets to deal with Nephara. Poor Trolleborg. Welcome to the World Cup.
Nephara closes out our schedule in the group stage. The #2 nation in the World used their great run in World Cup 84 to advance from being ranked in the teens to being where they are today. It is in the perfect spot on the schedule for both of our nations, because if we take care of business in our first two matches, then the finale is nothing more than playing for pride. We’ve only played Nephara once in our history in World Cup play, and that was a 3-1 win for the Cormorants. Like we said, take care of business against the first two opponents and we don’t need to worry about picking up a point against Nephara. Of course you want to win the group, but we’ll cross that bridge if we get to it.
Taking care of business starts against Trolleborg.
Will they do it?
See above. A lesson was taught last cycle. We'll see soon if they learned anything.
NS Sports’ only World Cup, World Bowl, World Cup of Hockey, World Baseball Classic and International Basketball Championships winner!
(Motorsports, college basketball, and volleyball, too)
Specific Titles: World Cup 50, 51; WBC 14, 16, 19 & 50; WB 8, 22, & 40; WCOH 11 & 39; IBC 13.
Also: CoH 39; Swamp Soccer 4, WVE (Vball) 6; NSCAA 3, 5 & 9; NSSCRA 7
Runner Up: CoH 40, WB 21, WcoH 8, IBC 12, WBC 13, 15, 47 & 48, DBC 21. 3rd Place @ WC 86
WC Qualified for: 45, 46, 49-61, 67, 79 (DNP WC 69-77), 81-86.
Games of the XIII Summer Olympiad: Second Most Medals
Hosted: World Cup 54, 67 & 84; CoH 57 & 73, BoF 47, CR 30, WB 16, WBC 18, 26, 40, 45 & 50, Some NSCAA, NSCH 1; WLC 7, 30 & 33.
Zwangzug
Issues Editor
Posts: 4593
Founded: Oct 19, 2006
Left-wing Utopia
*Rocket Boosters Go For Launch: Newmanistan-Zwangzug would be a big-ticket series in the World Baseball Classic; the Rockets and Zebras have contested three WBC finals as opponents, with Newmanistan leading the "season series" 2-1. Even their group-stage clash in WBC40, with an inexperienced Zebras team facing off against a Rockets side that would take third place overall, was captivating. But in football, it's a different story. Yes, the nations' histories have overlapped--they co-hosted Cup of Harmony 33 that ran in parallel with World Cup 41. But this is the first time they've simulataneously reached the last 32, much less faced each other there. Tim Marley's hustle and endurance are known to 1./ fans who have watched him with Rovers United, but beyond that, it's anybody's guess as to whether the home side will be inspired by the spirits of Tundra Falls or pressured by the expectations of their fans.
Brenecia and The Sarian are the other members of Group A; the latter association may ring a bell not so much for defeating Zwangzug in CoH 60 but rather for the FFI campaigns of CF Bondstad, who lost to Arlington City and Spenson Suburbia in consecutive finals.
*Squad Goals: Brandon Sonnabend has dismissed speculation that he would adopt a more attacking style at the proper as "ridiculous cynicism." "Look, it's nice to score goals," he admitted. "And it's especially nice to score goals on such a large stage, where fans from across the world are watching and maybe even recording your achievements on some kind of wiki thing. But not only would it be foolish to switch our tactics up just for that reason, it would also probably be bad for team cohesion." Sonnabend added that screwing around with a national team's strategy for the purpose of manipulating the "away goals" tiebreaker was "unsavory gamesmanship, but still a more reasonable idea than the former."
Commentators and writers from the notably-unstable region of Farfadillis have spearheaded the charge for more comprehensive encyclopedias, a charge eagerly taken up by the Arlington Collective. "I won't lie, Farfadillis isn't the first place I think of when I think of methodical record-keeping and analysis," admitted Cheryl Woen. "But it could be worse. It could be a lot worse." She shuddered in lieu of elaborating.
Of course, even defensive teams like Zwangzug and/or the Sarian can produce plenty of goals. Zwangzug, in fact, scored two against all-out-defenders Verceola, through Jaclyn Burkhart and Akhil Sibal, but were nevertheless held to a draw thanks to Fisher Ellery and Martin Bartoš' goals.
*Dispirited?: Johan Idmar claims he has "no concerns" about playing in Newmanistan, a nation whose religious tolerance isn't always extended to openly-atheist visitors. "I trust the hosts will do their best to welcome players and fans from across the world, and I'm not here to prosyletize or argue, I'm just trying to play my game. And if the Tundra Falls crowd is vocal, well, fans can be a big part of home-field advantage!" (Presumably in reference to the spirits said to support the home nation at the Proving Grounds.) "Besides, I'm going to be too busy to go on any silly TV shows."
At the allusion to Idmar's appearance on Interreligious Sniping, surprise guest Mikhail Struve insisted on weighing in on Group A. "Brenecia are the team to beat, no doubt," he opined, "Brelk-Xeral Erv's done some great things with the side, and Lauren Cheney is a very deserving Galactico candidate. But I think most people are underrating De Bondsaari. Both their formational flexibility and their stripes are very attractive football, and they should advance along with the Patriots." Struve refused to give further details, but based on what we know of the Divine Host of Archangels' doctrine, this isn't the sect's hottest take.
Akhil Sibal, for his part, looks forward to visiting a country with a different culture. "Yeah, I got a new car when I moved to Chernovets, it was great," he said. "But since there's only a handful of us who know how, I'm guessing I'll have to shuttle half the team around."
*"History Majors Can Go Suck It": It will come as no surprise to many that the 1./ (First Division) has followed its own path to the top eight of the IFCF coefficients. A few foreign fans, however, may be disappointed in that not only is it hard to watch our games overseas, Zwangzugians watching reputable streams of overseas leagues may be just as rare.
The aforementioned encyclopedic efforts not only chronicle the past; they also tabulate which television channels have purchased broadcast rights to future league play. Zwangzug, however, have made no inroads into these deals. "I mean, let's face it, our economy isn't...particularly good," admitted a league official. "Or free, or anything. Sure, we might not be an autarky, but at least in these matters we're being left behind by more prosperous and interconnected associations, and I don't see that changing any time soon."
It was cautiously proposed that a country with a large and sports-interested population like ours could take steps to remedy such problems, but this idea did not go over well. "Who do you think we are, another admirably-developed country?" challenged another administrator. "It says somewhere in the laws of ~realism and ~diversity that one cannot have even a somewhat affluent or fortunate populace without an oppressive backstory involving colonialism, imperialism, and/or all the terrible things that history masochists are into. And frankly, bleep that noise. If having our matches pirated by savvy foreigners and failing to profit at all by it is the price we have to pay for being mentally healthy and able to sleep at night, then we're going to bleeping pay it, and bleep you."
These outbursts are par for the course in the uneven world of Zwangzugian pedagogy. Several members of Parliament, who probably spend too much time on microblogging sites, were recently found to be reblatting messages to the effect of "we need to stop teaching the dangerous myth that zwangzug history began in 1965, or 2007, or 512, or indeed any time at all. history is dumb, let's just learn math."
Correction: this article originally mentioned Mikhail Struve without pointing out that he is pledged unto the Worshipful Cherubim. The Courier doesn't really regret the omission, but my editor told me I had to say something nice.
Factbook
IRC humor, (self-referential)
My issues
...using the lens of athletics to illustrate national culture, provide humor, interweave international affairs, and even incorporate mathematical theory...
WARNING: by construing meaning from this sequence of symbols, you have given implicit consent to the theory that words have noncircular semantic value and can be used to encode information about an external universe. Proceed with caution.
Commonwealth of Baker Park
Posts: 1614
Founded: Jan 10, 2018
Civil Rights Lovefest
Can Baker Park win the World Cup?
Scott McCrae
The Daily Mail National Soccer Writer
The Commonwealth National Team arrived on Sonnel, reaching their destination in Drawkland at the southeastern city of Shire-Port--one of two hosts in the Ellzidanet region (the other being Elstrund)-- where the NT have chosen to make their camp.
As they were for the qualifying draw, BP were the last nation drawn of the 32 teams, placing them in Group H along with Audioslavia, Kandorith and top seeds Turori; the coincidences of the draw have parallels with the third occasion when the Commonwealth were the final nation drawn at a World Cup, for the group stage of WC 83 in Equestria & Banija.
Of the opponents that will line up against the Black & Gold in Drawkland, only the Bulls have a history with the Bees; their lone matchup was at the Stadium of the Restoration in Busukuma in the final Group H contest in that previously mentioned Cup--a 3-0 BP victory that clinched a perfect record and a third successive advancement to the knockout rounds. Now, as then, the sides will meet in the group finale.
On matchday 1, 7th ranked Baker Park will go head to head with the 6th ranked Eels for the first time, and there is a connection to WC 83 that will probably not mean much in the bigger picture.
Turori were the second ranked side in the multiverse, having been the runners-up in World Cup 82, and the Commonwealth opened their fourth WCF against top ranked Vilita--the defending Champions--in Jinja City, at the BCEL Stadium with a shock 1-0 win. The Eels and Mriin were scheduled to play on MD 2 at the BCEL, which pulled the strings together; Turori based Cocoa-bo had entered into a licensing partnership with the Commonwealth's largest beverage producer on the eve of those Finals, which meant that the chocolately delectable was available for sale inside the stadium.
BP had faced off against the combined Vilita & Turori side in AOCAF LVIII in a famous 4-2 victory at Rogers Stadium in Endborough which propelled the hosts to the Semi-Final, so they will be looking to get the hat trick of victories over the Vilitan Cove conurbation.
The other side in Group H, Kandorith, can also be pulled into a tenuous connection to the WC83 storyline; appearing in the Finals after a long absence, The South Pacific nation is currently ranked 36 in the KPB, which is the same ranking that The Maccabees held in Group H when Baker Park faced them on matchday 2 at the Albert III Stadium in Dodoma.
The resulting 2-1 victory was memorable for the outpouring of emotion by the neutrals in the crowd of 48,000 towards the Commonwealth side in the wake of the recent State Visit that Kabaka Mwanga made to Baker Park, plus the fact that the NT kit for that match incorporated the Banijan flag motif on the sleeve trim.
On the topic of what the realistic expectations should be for this seventh straight appearance in the 32 team World Cup Finals, the answer seems to be obvious; having reached the semi-final while facing knockout matches against Starblaydia, Valanora, and eventual runner-up Nephara, then dropping the 3rd Place match to the current top ranked side Banija, there can be no question that this nations' finest group of players to wear the national team jersey should look upon anything short of a repeat appearance in the last 4 as a disappointment.
AOCAF LXII Champions
2x Under-18 World Cup (SWC 5&9) Champions
Baptism of Fire 67 Runner-Up
AOCAF LVIII (co-hosts), LX Third Place
World Cup 85 Fourth Place
World Cup 84 Co-hosts
World Cup 81/82/83/84/86 Round of 16
World Cup 80 Group Stage
AOBC 5 Champions
Football
NSCF 21 Mineral Conference Champions
playoff semi-finalists
NSCF 18 Mineral Conference Champions
playoff quarterfinalist
NSCF 19 & 20 Mineral Conference Champions
Lacrosse
WLC Vice President
WLC 34 Fourth Place
WLC 30/31(host)/32/33/35 Quarterfinal
WLC 29 Playoff Round
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2021-04-13 18:02:53
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http://crypto.stackexchange.com/questions/16589/rsa-encryption-and-signature-padding-flaw
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# RSA encryption and signature padding flaw
This is Oxford's computer security exam question:
Suppose that Bob has published an RSA encryption key $ke$ (retaining in secret the corresponding decryption key $kd$), and Alice has published a public RSA verification key $kv$ (retaining in secret the corresponding signing key $ks$). Alice wants to send an integrity protected and confidential message $m$ to Bob by transmitting the pair:
$E_{ke}(m)$, $SIGN_{ks}(E_{ke}(m))$
Bob will perform the obvious decryption and verification, rejecting messages which are incorrectly signed or have incorrect padding. However, Alice and Bob use defective padding schemes. The encryption process RSA encrypts the byte string ‘padding 00 message’, where "padding" contains random nonzero bytes, 00 indicates a single zero byte, and message is the intended message (which is assumed to fit within the RSA modulus). The signature process RSA-signs the byte string ‘01 01 message’ (and again the message is assumed to fit within the modulus in this format).
Greedy is an active attacker who intercepts Alice’s communication to Bob. He knows that the message contains a sum of money which is to be transferred by Bob (a bank) to Greedy himself. Show how Greedy can tamper with Alice’s message to give himself a non-negligible chance of getting very rich.
I have read RSA Encryption and Signature - Weak Padding but the attack found in that question can not control the content of $m$. Can anyone show me how to answer this question?
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Do Alice and Bob use the same modulus? $\;$ – Ricky Demer Jun 6 at 7:58
@RickyDemer : I think they are different, this is the exact original exam question. – user3283751 Jun 6 at 9:03
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2014-12-28 13:03:23
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https://www.clutchprep.com/chemistry/practice-problems/143920/what-is-the-percent-ionic-character-of-the-h-br-bond-express-your-answer-numeric
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# Problem: What is the percent ionic character of the H−Br bond?Express your answer numerically to three significant figures.The dipole moment for a bond is a measure of the magnitude of polarity (how polar the bond is). Dipole moments can be used to calculate percent ionic character.The dipole moment is symbolized by μ (Greek lowercase "mu") and is equal to the charge, Q, of the atoms (measured in Coulombs, C) times the distance, r, of separation between the atoms (measured in meters, m):μ = Q × rTherefore, the units of dipole moment are Coulomb-meters, C⋅m. Another unit of dipole moment is the debeye, D:1 D = 3.34 × 10−30 C⋅mThe percent ionic character is calculated by dividing the charge, Q, by the charge on a single electron, 1.60×10−19 C⋅m.
###### Problem Details
What is the percent ionic character of the H−Br bond?
The dipole moment for a bond is a measure of the magnitude of polarity (how polar the bond is). Dipole moments can be used to calculate percent ionic character.
The dipole moment is symbolized by μ (Greek lowercase "mu") and is equal to the charge, Q, of the atoms (measured in Coulombs, C) times the distance, r, of separation between the atoms (measured in meters, m):
μ = Q × r
Therefore, the units of dipole moment are Coulomb-meters, C⋅m. Another unit of dipole moment is the debeye, D:
1 D = 3.34 × 10−30 C⋅m
The percent ionic character is calculated by dividing the charge, Q, by the charge on a single electron, 1.60×10−19 C⋅m.
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2020-10-25 20:01:03
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https://www.physicsforums.com/threads/spherical-capacitor-discharging-through-radial-resistor.956196/
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# Spherical Capacitor Discharging Through Radial Resistor
## Homework Statement
A spherical capacitor has internal radius ##a## and external radius ##b##. At time ##t = 0##, the charge of the
capacitor is ##Q_0## Then the two shells are connected by a resistor in the radial direction of resistance ##R##. Find the Poynting vector and the energy dissipation in a small volume around the resistor and compare it to the time variation of the electrostatic energy.
## Homework Equations
##\begin{aligned} \nabla \times \vec{\mathbf{B}} - \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \mu_0 \vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = \frac{\rho}{\epsilon_0} \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}##
## The Attempt at a Solution
When the resistor is connected to the shells the system becomes an RC circuit, so the charge on the capacitor is now ##Q(t) = Q_0 \exp{(-t/\tau)}## with ##\tau =RC = \frac{4\pi R}{\frac{1}{a} -\frac{1}{b}}##. This results in a flow of current but is the first source of confusion for me. Does the current flow through the resistor connecting the shells? If so then we have a radial current ##\vec{\mathbf{j}} =-\frac{dQ}{dt} \vec{\hat{r}}##. The electric field between the shells is also changing as the charge decreases so there is a displacement current determined by ##\vec{\mathbf{E(t)}} = \frac{Q(t)}{r^2 \epsilon_0} \vec{\hat{r}}##. Using these I could find the magnetic field around the resistor from Maxwell's equations and then Poynting's vector by taking a cross product and then integrate for a surface surrounding the resistor to compare against the time derivative of the electrostatic energy ##U = \frac{Q(t)^2}{2C}##. But before proceeding further I want to make sure that I'm understanding the physics here properly.
nrqed
Homework Helper
Gold Member
## Homework Statement
A spherical capacitor has internal radius ##a## and external radius ##b##. At time ##t = 0##, the charge of the
capacitor is ##Q_0## Then the two shells are connected by a resistor in the radial direction of resistance ##R##. Find the Poynting vector and the energy dissipation in a small volume around the resistor and compare it to the time variation of the electrostatic energy.
## Homework Equations
##\begin{aligned} \nabla \times \vec{\mathbf{B}} - \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \mu_0 \vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = \frac{\rho}{\epsilon_0} \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}##
## The Attempt at a Solution
When the resistor is connected to the shells the system becomes an RC circuit, so the charge on the capacitor is now ##Q(t) = Q_0 \exp{(-t/\tau)}## with ##\tau =RC = \frac{4\pi R}{\frac{1}{a} -\frac{1}{b}}##.
Are you using units where ##\epsilon_0=1##? If not, you are missing some factors there.
To answer your question, yes, the current will be flowing through the resistor.
So then the rest of the solution would proceed `as I've outlined? Use the expression for curl in spherical coordinates, find which component of B is non-zero and then turn the crank to find Poynting's vector and compare with the time derivative of the electrostatic energy? Regarding units I'm missing ##\mu_0## and ##\epsilon_0## in the Ampere-Maxwell law.
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2022-07-05 13:10:53
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https://rforanalytics.com/14.1-logistic-regression.html
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## 14.1 Logistic Regression
• Logistic Regression belongs to the class of generalised linear models (glms)generalised linear models (glms)
• Used to model data with a dichotomous response variable.
• Logistic regression models the conditional probability of the response variable rather than its value.
• A logit link function, defined as $$logit\,p=log[p/(1-p)]$$, is used to transform the output of a linear regression to be suitable for probabilities.
• A linear model for these transformed probabilities can be setup as
$$$logit\,p=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots\beta_{k}x_{x} \tag{14.1}$$$
• R provides the glm function for modelling generalised linear models including the logistic regression model
• We will use the caret package to model logistic regression later in this topic.
• See and for further details on logistic regression.
### References
Boehmke, Brad, and Brandon M Greenwell. 2019. Hands-on Machine Learning with r. CRC Press. https://bradleyboehmke.github.io/HOML/.
Hastie, Trevor, Robert Tibshirani, Gareth James, and Daniela Witten. 2013. An Introduction to Statistical Learning with Applications in r. Springer New York.
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2022-08-16 01:15:45
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https://www.physicsforums.com/threads/conservation-of-linear-and-angular-momentum.231495/
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# Conservation of Linear and Angular Momentum
NeuronalMan
## Homework Statement
We have two spheres of equal mass, M, and radius R. One of them is initially at rest (in space, apparently) and the other is moving downwards with a velocity v0. Air resistance is given by an approximation (see below). Consider a completely inelastic, linear collision between the two spheres. What is the velocity v1 of the center of mass of the system of two spheres, after the collision? And similarly, what is the angular velocity omega1 of the sytem?
## Homework Equations
The only equation that I think might be relevant is that of air resistance, which is Fv = -kvV, where kv is a constant. This is to say that the air resistance is proportional to the velocity, though. Obviously other relevant equations would be conservation of linear and angular momentum, but these I consider to be so fundamental that I don't write them down.
## The Attempt at a Solution
In problems like these, one is inclined to use conservation of momentum, and that is what first crossed my thoughts when I looked at this problem. As for the angular momentum, the net external torque is zero (at least for the linear case), so angular momentum is conserved. That should make it easy to find the angular velocity omega1 for the system of two spheres. However, I have never seen any problem like this, concerning linear momentum. It seems to me that the linear momentum cannot be conserved during the collision, because the net force acting on the system, isn't zero. Namely the weight due to gravity and air resistance. Thus, I am left with no other way of finding the linear velocity v1.
There is also a general case in which the line between the centers of the spheres, form an angle, theta, with the horizontal. And in the general case, the same frustration arises for angular momentum, as I believe air resistance act with a torque on the system.
I didn't yet take the time to learn the LaTeX-codes for writing formulas and equations, but I hope you can bear with my symbols.
Last edited:
Homework Helper
Hi NeuronalMan,
It is true that the net force on the system is not zero if there is gravity and air resistance. However, if the time duration of the collision is very small, we can consider momentum to be conserved, becuase the momentum relation is (for average external force):
$$\vec F_{\rm ext} \Delta t = \Delta \vec p$$
So if $\Delta t$ is vanishingly small, then momentum is for practical purposes conserved over the time interval of the collision.
Lord Crc
I'm having an assignment with the exact same problem. In it there's a sentence saying that we can consider the collision to be instantaneous and thus we can ignore the gravity and air resistance.
NeuronalMan
Hi NeuronalMan,
It is true that the net force on the system is not zero if there is gravity and air resistance. However, if the time duration of the collision is very small, we can consider momentum to be conserved, becuase the momentum relation is (for average external force):
$$\vec F_{\rm ext} \Delta t = \Delta \vec p$$
So if $\Delta t$ is vanishingly small, then momentum is for practical purposes conserved over the time interval of the collision.
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2022-12-03 05:56:12
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http://terrytao.wordpress.com/2012/03/07/the-asymptotic-distribution-of-a-single-eigenvalue-gap-of-a-wigner-matrix/
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I’ve just uploaded to the arXiv my paper The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, submitted to Probability Theory and Related Fields. This paper (like several of my previous papers) is concerned with the asymptotic distribution of the eigenvalues ${\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)}$ of a random Wigner matrix ${M_n}$ in the limit ${n \rightarrow \infty}$, with a particular focus on matrices drawn from the Gaussian Unitary Ensemble (GUE). This paper is focused on the bulk of the spectrum, i.e. to eigenvalues ${\lambda_i(M_n)}$ with ${\delta n \leq i \leq (1-\delta) i n}$ for some fixed ${\delta>0}$.
The location of an individual eigenvalue ${\lambda_i(M_n)}$ is by now quite well understood. If we normalise the entries of the matrix ${M_n}$ to have mean zero and variance ${1}$, then in the asymptotic limit ${n \rightarrow \infty}$, the Wigner semicircle law tells us that with probability ${1-o(1)}$ one has
$\displaystyle \lambda_i(M_n) =\sqrt{n} u + o(\sqrt{n})$
where the classical location ${u = u_{i/n} \in [-2,2]}$ of the eigenvalue is given by the formula
$\displaystyle \int_{-2}^{u} \rho_{sc}(x)\ dx = \frac{i}{n}$
and the semicircular distribution ${\rho_{sc}(x)\ dx}$ is given by the formula
$\displaystyle \rho_{sc}(x) := \frac{1}{2\pi} (4-x^2)_+^{1/2}.$
Actually, one can improve the error term here from ${o(\sqrt{n})}$ to ${O( \log^{1/2+\epsilon} n)}$ for any ${\epsilon>0}$ (see this previous recent paper of Van and myself for more discussion of these sorts of estimates, sometimes known as eigenvalue rigidity estimates).
From the semicircle law (and the fundamental theorem of calculus), one expects the ${i^{th}}$ eigenvalue spacing ${\lambda_{i+1}(M_n)-\lambda_i(M_n)}$ to have an average size of ${\frac{1}{\sqrt{n} \rho_{sc}(u)}}$. It is thus natural to introduce the normalised eigenvalue spacing
$\displaystyle X_i := \frac{\lambda_{i+1}(M_n) - \lambda_i(M_n)}{1/\sqrt{n} \rho_{sc}(u)}$
and ask what the distribution of ${X_i}$ is.
As mentioned previously, we will focus on the bulk case ${\delta n \leq i\leq (1-\delta)n}$, and begin with the model case when ${M_n}$ is drawn from GUE. (In the edge case when ${i}$ is close to ${1}$ or to ${n}$, the distribution is given by the famous Tracy-Widom law.) Here, the distribution was almost (but as we shall see, not quite) worked out by Gaudin and Mehta. By using the theory of determinantal processes, they were able to compute a quantity closely related to ${X_i}$, namely the probability
$\displaystyle {\bf P}( N_{[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]} = 0) \ \ \ \ \ (1)$
that an interval ${[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]}$ near ${\sqrt{n} u}$ of length comparable to the expected eigenvalue spacing ${1/\sqrt{n} \rho_{sc}(u)}$ is devoid of eigenvalues. For ${u}$ in the bulk and fixed ${x,y}$, they showed that this probability is equal to
$\displaystyle \det( 1 - 1_{[x,y]} P 1_{[x,y]} ) + o(1),$
where ${P}$ is the Dyson projection
$\displaystyle P f(x) = \int_{\bf R} \frac{\sin(\pi(x-y))}{\pi(x-y)} f(y)\ dy$
to Fourier modes in ${[-1/2,1/2]}$, and ${\det}$ is the Fredholm determinant. As shown by Jimbo, Miwa, Tetsuji, Mori, and Sato, this determinant can also be expressed in terms of a solution to a Painleve V ODE, though we will not need this fact here. In view of this asymptotic and some standard integration by parts manipulations, it becomes plausible to propose that ${X_i}$ will be asymptotically distributed according to the Gaudin-Mehta distribution ${p(x)\ dx}$, where
$\displaystyle p(x) := \frac{d^2}{dx^2} \det( 1 - 1_{[0,x]} P 1_{[0,x]} ).$
A reasonably accurate approximation for ${p}$ is given by the Wigner surmise ${p(x) \approx \frac{1}{2} \pi x e^{-\pi x^2/4}}$, which was presciently proposed by Wigner as early as 1957; it is exact for ${n=2}$ but not in the asymptotic limit ${n \rightarrow \infty}$.
Unfortunately, when one tries to make this argument rigorous, one finds that the asymptotic for (1) does not control a single gap ${X_i}$, but rather an ensemble of gaps ${X_i}$, where ${i}$ is drawn from an interval ${[i_0 - L, i_0 + L]}$ of some moderate size ${L}$ (e.g. ${L = \log n}$); see for instance this paper of Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou for a more precise formalisation of this statement (which is phrased slightly differently, in which one samples all gaps inside a fixed window of spectrum, rather than inside a fixed range of eigenvalue indices ${i}$). (This result is stated for GUE, but can be extended to other Wigner ensembles by the Four Moment Theorem, at least if one assumes a moment matching condition; see this previous paper with Van Vu for details. The moment condition can in fact be removed, as was done in this subsequent paper with Erdos, Ramirez, Schlein, Vu, and Yau.)
The problem is that when one specifies a given window of spectrum such as ${[\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)}, \sqrt{n} u + \frac{y}{\sqrt{n} \rho_{sc}(u)}]}$, one cannot quite pin down in advance which eigenvalues ${\lambda_i(M_n)}$ are going to lie to the left or right of this window; even with the strongest eigenvalue rigidity results available, there is a natural uncertainty of ${\sqrt{\log n}}$ or so in the ${i}$ index (as can be quantified quite precisely by this central limit theorem of Gustavsson).
The main difficulty here is that there could potentially be some strange coupling between the event (1) of an interval being devoid of eigenvalues, and the number ${N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)}$ of eigenvalues to the left of that interval. For instance, one could conceive of a possible scenario in which the interval in (1) tends to have many eigenvalues when ${N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)}$ is even, but very few when ${N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)}$ is odd. In this sort of situation, the gaps ${X_i}$ may have different behaviour for even ${i}$ than for odd ${i}$, and such anomalies would not be picked up in the averaged statistics in which ${i}$ is allowed to range over some moderately large interval.
The main result of the current paper is that these anomalies do not actually occur, and that all of the eigenvalue gaps ${X_i}$ in the bulk are asymptotically governed by the Gaudin-Mehta law without the need for averaging in the ${i}$ parameter. Again, this is shown first for GUE, and then extended to other Wigner matrices obeying a matching moment condition using the Four Moment Theorem. (It is likely that the moment matching condition can be removed here, but I was unable to achieve this, despite all the recent advances in establishing universality of local spectral statistics for Wigner matrices, mainly because the universality results in the literature are more focused on specific energy levels ${u}$ than on specific eigenvalue indices ${i}$. To make matters worse, in some cases universality is currently known only after an additional averaging in the energy parameter.)
The main task in the proof is to show that the random variable ${N_{(-\infty,\sqrt{n} u + \frac{x}{\sqrt{n} \rho_{sc}(u)})}(M_n)}$ is largely decoupled from the event in (1) when ${M_n}$ is drawn from GUE. To do this we use some of the theory of determinantal processes, and in particular the nice fact that when one conditions a determinantal process to the event that a certain spatial region (such as an interval) contains no points of the process, then one obtains a new determinantal process (with a kernel that is closely related to the original kernel). The main task is then to obtain a sufficiently good control on the distance between the new determinantal kernel and the old one, which we do by some functional-analytic considerations involving the manipulation of norms of operators (and specifically, the operator norm, Hilbert-Schmidt norm, and nuclear norm). Amusingly, the Fredholm alternative makes a key appearance, as I end up having to invert a compact perturbation of the identity at one point (specifically, I need to invert ${1 - 1_{[x,y]}P1_{[x,y]}}$, where ${P}$ is the Dyson projection and ${[x,y]}$ is an interval). As such, the bounds in my paper become ineffective, though I am sure that with more work one can invert this particular perturbation of the identity by hand, without the need to invoke the Fredholm alternative.
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2013-12-12 11:47:51
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https://abhyasonline.in/questions/Maths/Constructions/If%20the%20angles%20of%20a%20triangle%20are%20in%20the%20ratio%201%20%202%20%207%20then%20the%20triangle%20is%20_____________/
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Maths / Constructions / Construction of Triangle using ASA Criteria
QUESTION
If the angles of a triangle are in the ratio 1 : 2 : 7 then the triangle is _____________
OPTIONS A. Acute angled B. Obtuse angled C. Right angled D. Right angled isosceles
Right Option : B
EXPLANATION
Explain TypeExplanation Content
Text
Let the angles be 1x, 2x, 7x = $180^0$
1x + 2x + 7x = 180
10x = 180
x = 18
The angles are : 18, 36, 126. It is obtuse angled triangle.
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2021-08-02 15:58:07
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https://gomathanswerkey.com/texas-go-math-grade-3-lesson-9-5-answer-key/
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# Texas Go Math Grade 3 Lesson 9.5 Answer Key Multiply 2-Digit by 1-Digit Numbers
Refer to our Texas Go Math Grade 3 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 3 Lesson 9.5 Answer Key Multiply 2-Digit by 1-Digit Numbers.
## Texas Go Math Grade 3 Lesson 9.5 Answer Key Multiply 2-Digit by 1-Digit Numbers
Essential Question
How can you use strategies to multiply a 2-digit number by a 1-digit number?
Multiply the 1 digits together first. Cross-multiply by multiplying diagonal digits, then add the two sums together. Multiply the 2-digit numbers.
Explanation:
Here is one way that you can speed up the process of 2-digit by 1-digit multiplication:
Multiply the 1 digits together first. Cross-multiply by multiplying diagonal digits, then add the two sums together. Multiply the 2-digit numbers.
Unlock the Problem
A Thoroughbred racehorse can run at speeds of up to 60 feet per second. During practice, Celia’s horse runs at a speed of 36 feet per second. How far does her horse run in 3 seconds?
• Underline important information.
• Is there information you will not use? If so, cross out the information.
Example 1.
Multiply. 3 × 36
So, Celia’s racehorse runs ___ feet in 3 seconds.
So, Celia’s racehorse runs 108 feet in 3 seconds.
Explanation:
Number of feet Celia’s horse runs per second = 36.
Distance her horse runs in 3 seconds = 3 × Number of feet Celia’s horse runs per second
= 3 × 36
= (3 × 10) + (3 × 10) + (3 × 10) + (3 × 6)
= 30 + 30 + 30 + 18
= 60 + 30 + 18
= 90 + 18
= 108 feet.
Math Talk
Mathematical Processes
Look at Step 1. Explain how the blocks show the regrouping of the 18 ones.
Example 2.
Multiply. 8 × 22
So, 8 × 22 = ____.
Math talk
Mathematical Processes
Why is 8 × 2 tens recorded as 160 and not as 16?
8 × 2 tens is recorded as 160 and not as 16 because 8 × 2 tens = 8 × 20 = 160 not 16.
Explanation:
8 × 2 tens = 8 × (2 × 10)
= 8 × 20
Multiply tens and ones place.
= 160.
Share and Show
Question 1.
Use the model to find the product.
Explanation:
2 × 36 = (2 × 3 tens) + (2 × 6 ones)
= [2 ×(3 × 10)]+ [2 × (6 × 1)]
= (2 × 30) + (2 × 6)
= 60 + 12
= 72.
Find the product. Tell which strategy you used.
Question 2.
Strategy used is multiplication and addition to find the product.
Explanation:
42 × 4 = (40 × 4) + (2 × 4)
= 160 + 8
= 168.
Question 3.
Strategy used is multiplication and addition to find the product.
Explanation:
32 × 6 = (30 × 6) + (2 × 6)
= 180 + 12
= 192.
Question 4.
Strategy used is multiplication and addition to find the product.
Explanation:
85 × 3 = (80 × 3) + (5 × 3)
= 240 + 15
= 255.
Question 5.
Strategy used is multiplication and addition to find the product.
Explanation:
$63 × 7 = ($60 × 7) + ($3 × 7) =$420 + $21 =$441.
Math Talk
Mathematical Processes
Describe the steps for using place value and regrouping to find 3 × 78.
Steps for using place value and regrouping to find 3 × 78:
1. Multiply the tens place and record the value.
2. Multiply the ones place and record the value.
3. Do add the both vales.
3 × 78 = 234.
Explanation:
Steps for using place value and regrouping to find 3 × 78:
1. Multiply the tens place and record the value:
3 × 70 = 210.
2. Multiply the ones place and record the value:
3 × 8 = 24.
3. Do add the both vales:
=> (3 × 70) + (3 × 8)
= 210 + 24
= 234.
Problem Solving
Use the table for 6-8.
Question 6.
How far can a desert cottontail run in 9 seconds?
Distance desert cottontail runs 9 seconds = 198 feet.
Explanation:
Distance desert cottontail runs per second = 22 feet.
Distance desert cottontail runs 9 seconds = 9 × Distance desert cottontail runs per second
= 9 × 22
= (9 × 10) + (9 × 10) + (9 × 2)
= 90 + 90 + 18
= 180 + 18
= 198 feet.
Question 7.
Analyze A black-tailed jackrabbit hops about 7 feet in a single hop. How far can it hop in 5 seconds?
Distance black-tailed jackrabbit hops in a 5 seconds = 35 feet.
Explanation:
Distance black-tailed jackrabbit hops in a single hop per second = 7 feet.
Distance black-tailed jackrabbit hops in a 5 seconds = 5 × Distance black-tailed jackrabbit hops in a single hop per second
= 5 × 7
= 35 feet.
Question 8.
H.O.T. Multi-Step At the speeds shown, how much farther could a black-tailed jackrabbit run than a desert cottontail in 7 seconds?
203 feet farther a black-tailed jackrabbit run than a desert cottontail in 7 seconds.
Explanation:
Distance a black-tailed jackrabbit run in a second = 51.
Distance a black-tailed jackrabbit run in 7 seconds = 7 × Distance a black-tailed jackrabbit run in a second
= 7 × 51
= (7 × 10) + (7 × 10) + (7 × 10) + (7 × 10) + (7 × 10) + (7 × 1)
= 70 + 70 + 70 + 70 + 70 + 7
= 140 + 70 + 70 + 70 + 7
= 210 + 70 + 70 + 7
= 280 + 70 + 7
= 350 + 7
= 357 feet.
Distance a desert cottontail run in a second = 22.
Distance a desert cottontail run in 7 seconds = 7 × Distance a desert cottontail run in a second
= 7 × 22
=(7 × 10) + (7 × 10) +(7 × 2)
= 70 + 70 + 14
= 140 + 14
= 154 feet.
Difference:
Distance a black-tailed jackrabbit run in 7 seconds – Distance a desert cottontail run in 7 seconds
= 357 – 154
= 203 feet.
Question 9.
Multi-Step Mr. Wright bought a 3-pound bag of cat food and a 5 pound bag of dog food, There are 16 ounces in each pound. How many ounces of pet food did Mr. Wright buy?
Number of ounces of pet food Mr. Wright buy = 128.
Explanation:
Number of pounds bag of cat food Mr. Wright bought = 3.
Number of pounds bag of dog food Mr. Wright bought = 5.
Number of ounces in each pound = 16.
Number of ounces are in bag of cat food Mr. Wright bought = Number of pounds bag of cat food Mr. Wright bought ×Number of ounces in each pound
= 3 × 16
=(3 × 10 + (3 × 6)
= 30 + 18
= 48.
Number of ounces are in bag of dog food Mr. Wright bought = Number of pounds bag of dog food Mr. Wright bought × Number of ounces in each pound
= 5 × 16
=(5 × 10) + (5 × 6)
= 50 + 30
= 80.
Number of ounces of pet food Mr. Wright buy = Number of ounces are in bag of cat food Mr. Wright bought + Number of ounces are in bag of dog food Mr. Wright bought
= 48 + 80
= 128.
Question 10.
H.O.T. Reasoning The sum of two numbers is 31. The product of the two numbers is 150. What are the numbers?
The numbers are 6 and 25.
Explanation:
The sum of two numbers is 31.
The product of the two numbers is 150.
Let the numbers be X and Y.
Multiples of 150:
3 × 50 = 150. => 3 + 50 = 53.
5 × 30 = 150. => 5 + 30 = 35.
6 × 25 = 150. => 6 + 25 = 31.
10 × 15 = 150. => 10 + 15 = 25.
Question 11.
Write Math 6 × 87 is greater than 5 × 87. How much greater? Explain how you know without multiplying.
Difference = 87.
Without multiply:
(6 × 87) – (5 × 87) = difference of multiple (6 – 5) × 87 = 1 × 87 = 87.
Explanation:
Product of 6 × 87 = (6 × 10) + ( 6 × 10) + (6 × 10) + ( 6 × 10) + (6 × 10) + ( 6 × 10) + (6 × 10) + ( 6 × 10) + (6 × 7)
= 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 42
= 120 + 60 + 60 + 60 + 60 + 60 + 60 + 42
= 180 + 60 + 60 + 60 + 60 + 60 + 42
= 240 + 60 + 60 + 60 + 60 + 42
= 300 + 60 + 60 + 60 + 42
= 360 + 60 + 60 + 42
= 420 + 60 + 42
= 480 + 42
= 522.
Product of 5 × 87 = (5 × 10) + (5 × 10) + (5 × 10) + (5 × 10) + (5 × 10) + (5 × 10) + (5 × 10) + (5 × 10) + (5 × 7)
= 50 + 50 + 50 + 50 + 50 + 50 + 50 + 50 + 35
= 100 + 50 + 50 + 50 + 50 + 50 + 50 + 35
= 150 + 50 + 50 + 50 + 50 + 50 + 35
= 200 + 50 + 50 + 50 + 50 + 35
= 250 + 50 + 50 + 50 + 35
= 300 + 50 + 50 + 35
= 350 + 50 + 35
= 400 + 35
= 435.
Difference:
Product of 6 × 87 – Product of 5 × 87
= 522 – 435
= 87.
Without multiply:
(6 × 87) – (5 × 87) = difference of multiple (6 – 5) × 87 = 1 × 87 = 87.
Fill in the bubble for the correct answer. Use a strategy to solve.
Question 12.
Clayton bought 7 boxes of markers for an art project. There are 12 markers in each box. How many markers did he buy?
(A) 84
(B) 14
(C) 19
(D) 74
Number of markers he buy = 84.
(A) 84.
Explanation:
Number of boxes of markers for an art project Clayton bought = 7.
Number of markers in each box = 12.
Number of markers he buy = Number of boxes of markers for an art project Clayton bought × Number of markers in each box
= 7 × 12
= (7 × 10) + ( 7 × 2)
= 70 + 14
= 84.
Question 13.
Apply Alana has been on the swim team for 3 years. She practices 26 weeks a year. How many weeks has she practiced?
(A) 68
(B) 618
(C) 78
(D) 6,018
Number of weeks she practiced = 78.
(C) 78.
Explanation:
Number of years Alana has been on the swim team = 3.
Number of weeks a year she practices = 26.
Number of weeks she practiced = Number of years Alana has been on the swim team × Number of weeks a year she practices
= 3 × 26
= (3 × 10) + (3 × 10) +(3 × 6)
= 30 + 30 + 18
= 60 + 18
= 78.
Question 14.
Multi-Step At Huff and Puff Day Care, the art teachers have 5 sets of 16 paints for morning classes and 4 sets of 12 paints for afternoon classes. How many paints do the art teachers have?
(A) 80
(B) 128
(C) 48
(D) 20
Total number of paints the art teachers have = 128.
(B) 128.
Explanation:
Number of sets of paints the art teachers have for morning classes= 5.
Number of paints for morning classes = 16.
Total Number of paints the art teachers have for morning classes = Number of sets of paints the art teachers have for morning classes × Number of paints for morning classes
= 5 × 16
= (5 × 10) + (5 × 6)
= 50 + 30
= 80.
Number of sets of paints the art teachers have for afternoon classes = 4.
Number of paints for afternoon classes = 12.
Total Number of paints the art teachers have for afternoon classes = Number of sets of paints the art teachers have for afternoon classes × Number of paints for afternoon classes
= 4 × 12
= (4 × 10) + (4 × 2)
= 40 + 8
= 48.
Total number of paints the art teachers have = Total Number of paints the art teachers have for morning classes + Total Number of paints the art teachers have for afternoon classes
= 80 + 48
= 128.
Texas Test Prep
Question 15.
Mrs. Sawyer bought a book for $25 and 3 toys for$13 each. How much change should she get back from a $100 bill? (A)$46
(B) $36 (C)$26
(D) $64 Answer: Amount of change Mrs. Sawyer gets back =$36.
(B) $36. Explanation: Cost of a book Mrs. Sawyer bought =$25.
Cost of a toy Mrs. Sawyer bought each = $13. Number of toys Mrs. Sawyer bought = 3. Amount of bill =$100.
Total cost of toys Mrs. Sawyer bought = Cost of a toy Mrs. Sawyer bought each × Number of toys Mrs. Sawyer bought
= $13 × 3 = ($10 × 3) + ($3 × 3) =$30 + $9 =$39.
Total amount of a book and 3 toys Mrs. Sawyer bought = Cost of a book Mrs. Sawyer bought + Total cost of toys Mrs. Sawyer bought
= $25 +$39
= $64. Amount of change Mrs. Sawyer gets back = Amount of bill – Total amount of a book and 3 toys Mrs. Sawyer bought =$100 – $64 =$36.
### Texas Go Math Grade 3 Lesson 9.5 Homework and Practice Answer Key
Find the product. Tell which strategy you used.
Question 1.
Explanation:
Strategy used is multiplication and addition to find the product.
53 × 4 = (50 × 4) + (3 × 4)
= 200 + 12
= 212.
Question 2.
Strategy used is multiplication and addition to find the product.
Explanation:
64 × 6 = (60 × 6) + (4 × 6)
= 360 + 24
= 384.
Question 13.
Strategy used is multiplication and addition to find the product.
Explanation:
47 × 9 = (40 × 9) + (7 × 9)
= 360 + 63
= 423.
Question 14.
Strategy used is multiplication and addition to find the product.
Explanation:
82 × 3 = (80 × 3) + (2 × 3)
= 240 + 6
= 246.
Problem Solving
Use the table for 5-7.
Question 5.
The mayor of a town buys 4 packages of balloons for a street festival. How many balloons did the mayor buy?
Total number of balloons the mayor buy = 232.
Explanation:
Number of packages of balloons for a street festival the mayor of a town buys = 4.
Number of balloons in each package = 58.
Total number of balloons the mayor buy = Number of packages of balloons for a street festival the mayor of a town buys × Number of balloons in each package
= 4 × 58
= (4 × 50) + ( 4 × 8)
= 200 + 32
= 232.
Question 6.
Roberto buys 4 packages of noisemakers and 3 packages of hats. Does he have more hats or more noisemakers? Explain.
He has 7 more hats than noisemakers.
Explanation:
Number of packages of noisemakers Roberto buys = 4.
Number of noisemakers in each package = 17.
Total number of noisemakers Roberto buys = Number of packages of noisemakers Roberto buys × Number of noisemakers in each package
= 4 × 17
= (4 × 10) + (4 × 7)
= 40 + 28
= 68.
Number of packages of hats Roberto buys = 3.
Number of hats in each package = 25.
Total number of hats Roberto buys = Number of packages of hats Roberto buys × Number of hats in each package
= 3 × 25
= (3 × 20) + (3 × 5)
= 60 + 15
= 75.
Difference:
Total number of hats Roberto buys – Total number of noisemakers Roberto buys
= 75 – 68
= 7.
Question 7.
Nan buys 3 packages of party favors, If she has 100 people coming to a party, does she have enough favors? Explain.
No, she will not have enough favors to give to the people coming to a party.
Explanation:
Number of packages of party favors Nan buys = 3.
Number of favors in each package = 32.
Total number of favors Nan buys = Number of packages of party favors Nan buys × Number of favors in each package
= 3 × 32
= (3 × 30) + (3 × 2)
= 90 + 6
= 96.
Number of people coming to a party = 100.
Difference:
Number of people coming to a party – Total number of favors Nan buys
= 100 – 96
= 4.
Question 8.
Erika has 6 boxes of buttons. Each box holds 28 buttons. Explain how Erika can use place value and regrouping to find how many buttons there are.
Total number of buttons Erika has = 168.
Explanation:
Number of boxes of buttons Erika has = 6.
Number of buttons each box holds = 28.
Total number of buttons Erika has = Number of boxes of buttons Erika has × Number of buttons each box holds
= 6 × 28
Multiply 6 with the place value of 2 = 2 tens (2 ×10 = 20) and add to place value of 8 = ones (8 ×1 = 8)
= (6 × 20) + (6 × 8)
= 120 + 48
= 168.
Lesson Check
Texas Test Prep
Question 9.
It takes a train 42 minutes to ride ‘from Coltsville to Newburgh. If the train makes 8 trips in one day, how many minutes does it travel?
(A) 336 minutes
(B) 236 minutes
(C) 322 minutes
(D) 412 minutes
Number of minutes it travels = 336.
(A) 336 minutes.
Explanation:
Number of minutes a train takes to ride ‘from Coltsville to Newburgh = 42.
Number of trips a train takes in one day = 8.
Number of minutes it travels = Number of minutes a train takes to ride ‘from Coltsville to Newburgh × Number of trips a train takes in one day
= 42 × 8
= (40 × 8) + (2 × 8)
= 320 + 16
= 336.
Question 10.
Charlie puts together boxes of model cars at a factory. If he packs 18 boxes in one hour, how many boxes does he pack in 7 hours?
(A) 187
(B) 156
(C) 76
(D) 126
Number of boxes he pack in 7 hours = 126.
(D) 126.
Explanation:
Number of boxes he packs in one hour = 18.
Number of boxes he pack in 7 hours = 7 × Number of boxes he packs in one hour
= 7 × 18
= (7 × 10) + (7 × 8)
= 70 + 56
= 126.
Question 11.
A bookstore has 6 shelves of art books. There are 27 books on each shelf. how many art hooks are in the bookstore?
(A) 33
(B) 44
(C) 112
(D) 162
Number of art books are in the bookstore = 162.
(D) 162.
Explanation:
Number of shelves of art books a bookstore has = 6.
Number of books on each shelf = 27.
Number of art books are in the bookstore = Number of shelves of art books a bookstore has × Number of books on each shelf
= 6 × 27
= (6 × 20) + (6 × 7)
= 120 + 42
= 162.
Question 12.
A deli has 6 boxes of soup cans. Each box holds 14 cans. How many soup cans does the deli have?
(A) 64
(B) 20
(C) 84
(D) 24
Number of soup cans the deli have = 84.
(C) 84.
Explanation:
Number of boxes of soup cans a deli has = 6.
Number of cans each box holds = 14.
Number of soup cans the deli have = Number of boxes of soup cans a deli has × Number of cans each box holds
= 6 × 14
= (6 × 10) + (6 × 4)
= 60 + 24
= 84.
Question 13.
Multi-Step Ashley has 4 bags each of red marbles and yellow marbles. Each bag holds 14 marbles. How many marbles does Ashley have?
(A) 56
(B) 112
(C) 39
(D) 92
Number of marbles Ashley have = 112.
(B) 112.
Explanation:
Number of bags of each red marbles and yellow marbles Ashley has = 4.
Number of marbles each bag holds = 14.
Number of marbles Ashley have = (2 × Number of bags of each red marbles and yellow marbles Ashley has ) × Number of marbles each bag holds
= (2 × 4) × 14
= 8 × 14
= (8 × 10) + (8 × 4)
= 80 + 32
= 112.
Question 14.
Multi-Step Devon collects 23 bottles and 4 boxes of 16 cans on recycling day. His goal is to collect 100 items. How many more items does Devon need to reach his goal?
(A) 43
(B) 39
(C) 87
(D) 13
Number of more items Devon needs to reach his goal = 13.
(D) 13.
Explanation:
Number of bottles Devon collects = 23.
Number of boxes Devon collects = 4.
Number of cans he collects = 16.
Number of items to collect his goal = 100.
Number of items he collected on recycling day = Number of bottles Devon collects + (Number of boxes Devon collects × Number of cans he collects)
= 23 + (4 × 16)
= 23 + 64
= 87.
Number of more items Devon needs to reach his goal = Number of items to collect his goal – Number of items he collected on recycling day
= 100 – 87
= 13.
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2022-08-16 07:03:52
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|
https://ch.gateoverflow.in/17/gate-chemical-2018-question-7
|
Economy of evaporators used for concentrating sugarcane juice is
1. $\frac{kg\:of\:concentrated\:juice\:produced}{kg\:of\:steam\:supplied}$
2. $\frac{kg\:of\:steam\:supplied}{kg\:of\:sugarcane\:juice\:fed}$
3. $\frac{kg\:of\:water\:vaporized}{kg\:of\:steam\:supplied}$
4. $\frac{kg\:of\:sugarcane\:juice\:fed}{kg\:of\:water\:vaporized}$
|
2022-09-29 02:16:31
|
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|
https://aas.org/archives/BAAS/v25n4/aas183/abs/S5801.html
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Discovery of a Young Stellar-Wind Shell in Cygnus X
Session 58 -- Outflows and the Low Mass IMF
Oral presentation, Thursday, January 13, 10:15-11:45, Salon V Room (Crystal Gateway)
## [58.01] Discovery of a Young Stellar-Wind Shell in Cygnus X
L.A. Higgs (NRC/DRAO), H.J. Wendker (Hamburg), T.L. Landecker (NRC/DRAO) ()
A small ring-like radio source (G 79.29+0.46) has been discovered in the Cygnus X region from observations made with the DRAO Synthesis Telescope. The shell, $\sim 4^\prime$ in diameter, has a flux density of $\sim 550$ mJy at 1.42 GHz. Observations at three frequencies (1.49, 4.86 and 8.44 GHz) with the VLA have confirmed that the shell source is thermal in nature. A central radio source with a radio spectrum indicative of mass outflow was also detected. A heavily reddened optical counterpart of the latter has been found. A quick-look optical spectrum of this star (m$_R \sim 15.4^m$) in the 550 - 750 nm region shows only a heavily reddened continuum and a strong H$\alpha$ emission line. Extended wings of this line correspond to a wind velocity of $\sim 1400$ km s$^{-1}$. Observations at DRAO of the 21-cm line of atomic hydrogen indicate accelerated atomic gas associated with the ionized shell, and a shell expansion velocity of $\sim 70$ km s$^{-1}$. IRAS infrared data show the presence of warm dust ($T \sim 65$ K) in the ionized shell, and hotter dust ($T > 125$ K) associated with the central star and its wind. A consideration of all the observational data suggests a very luminous central star ($M_b \sim -11^m$) at a distance of $\sim 2$ kpc, having an effective temperature of $\sim 25,000$ K. It is suggested that it may be a member of the Luminous Blue Variable class. The age of the shell, having a mass of $\sim 15$ M$_\odot$, is estimated to be $\sim 10^4$ yr.
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2016-05-02 13:14:16
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http://pacechallenge.org/2019/vc/
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# PACE 2019 (Track Vertex Cover Exact)
## Vertex Cover
Given: A graph.
Task: Output a vertex cover of smallest size.
#### What is a vertex cover?
An undirected graph or simply a graph is a pair $G=(V,E)$ where $V\neq \emptyset$ is a set of vertices and $E \subseteq \{ \{u, v\} \subseteq V : u \neq v \}$ is a set of edges.
A vertex cover of a graph $G=(V,E)$ is a set $S\subseteq V$ such that for every edge $\{u,v\} \in E$ we have $\{u,v\} \cap S \neq \emptyset$.
See Details
1a. Exact Track
### Literature
[ChenKanjXia06]: Chen J., Kanj I.A., Xia G. Improved Parameterized Upper Bounds for Vertex Cover. In: Královič R., Urzyczyn P. (eds) Mathematical Foundations of Computer Science (MFCS’06). Lecture Notes in Computer Science, vol 4162. Springer. 2006.
[CyganEtAl15]: Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh: Parameterized Algorithms. ISBN 978-3-319-35702-7. Theoretical Computer Science. Springer. 2015.
[DonweyFellows13]: Rod Downey, Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer. 2013.
[Niedermeier06]: Rolf Niedermeier.
Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics And Its Applications. Oxford University Press. 2006.
[Wikipedia]: Wikipedia contributors, “Vertex cover,” Wikipedia, The Free Encyclopedia. (accessed November 28, 2018).
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2020-01-28 11:07:31
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http://scalability.org/?p=5839
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# Putting a 60 bay JackRabbit through some basic tests
Basic (conservative) configuration of the day jobs’ high performance tightly coupled storage system, no SSDs (apart from the OS drives). RAID6 LUNs, no RAID0’s. This is spinning rust folks. Nothing but spinning rust. In a realistic configuration.
And no, we haven’t yet begun to tune this.
Streaming writes, 1 thread per LUN:
Run status group 0 (all jobs):
WRITE: io=1279.5GB, aggrb=5944.6MB/s, minb=5944.6MB/s, maxb=5944.6MB/s, mint=220405msec, maxt=220405msec
5.9 GB/s sustained writes for this case. Writing 1.3 TB of data. In 220s. In a single 4U rack space. 160TB of usable space (out of 240TB in the unit). Lots of hot spares (so it detracts from the overall performance).
Run status group 0 (all jobs):
READ: io=1279.5GB, aggrb=5979.3MB/s, minb=5979.3MB/s, maxb=5979.3MB/s, mint=219116msec, maxt=219116msec
This is a respectable 5.9 GB/s sustained reads.
I should also point out that we are using effectively 40 data drives for this (60 drives – (2x hot spare + 2x parity drives/LUN) * 5 LUNs = 40 effective data drives). Call it 6GB/s for 40 drives, and we are averaging about 150MB/s per drive. Which is within a small delta of the overall per unit bandwidth of the drives … very nearly 160 MB/s for these. Call it about 94% bandwidth efficiency per drive.
Viewed 57892 times by 4011 viewers
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2014-10-23 02:15:39
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https://math.stackexchange.com/questions/473136/in-every-bipartite-graph-g-there-exists-a-vertex-v-such-that-v-is-matched-in-ev
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# In every bipartite graph G, there exists a vertex v such that v is matched in every maximum matching
As a part of König's Theorem proof I have to prove following lemma:
In every bipartite graph G, there exists a vertex v such that v is matched in every maximum matching.
Any help?
• I find your question hard to understand. Could you reformulate it (e.g. for every bipartite graph $G$, there exists vertex cover $C \subset G$ such that...)? – dtldarek Aug 22 '13 at 7:45
• Do you mean that in every bipartite graph $G,$ there exists a vertex $v$ such that $v$ is matched in every maximum matching? – Will Orrick Aug 22 '13 at 17:49
• @WillOrrick Yes, exactly. – alop789312 Aug 23 '13 at 21:48
This answer is not very direct (in that it uses other results) so I'm not sure how useful it will be to you.
By a theorem of Gallai's, a graph is factor critical if it is connected and for every vertex $v$, there exists a vertex which misses $v$.
Theorem (Gallai): Let $G(V,E)$ be a connected graph. $G$ is factor critical if for every vertex $v\in V$, there exists a maximum matching which misses $v$.
Proof: We shall prove that if $G$ is a graph such that for every vertex $v\in V$, there exists a maximum matching which misses $v$, then every maximum matching of $G$ misses exactly one vertex, i.e. each maximum matching is a near-perfect matching.
Suppose for the sake of contradiction that there exists a maximum matching which misses two (or more) vertices of $G$. It follows that every maximum matching misses two (or more vertices). Let $M$ be the maximum matching with missed vertices $u$ and $v$ separated by the smallest distance over all other matchings. Then $u$ and $v$ cannot be adjacent since otherwise we may add edge $(u,v)$ to $M$.
Let $w$ be an intermediate vertex on the shortest $uv$-path. Let $N$ be the matching which misses $w$ such that $|N\cap M|$ is maximal. $N$ cannot miss $u$ or $v$ since otherwise, $wu$ or $wv$ would be a shorter path between missed vertices, contradicting the minimality of $u$ and $v$ in $M$. It follows that $N$ must miss some vertex $x\neq w$ which is covered by $M$ (recall that they cover the same number of vertices). Let $y$ be the vertex such that $(x,y)\in M$. There must exist an edge $e$ adjacent to $y$ in $N$ for otherwise $x$ and $y$ would be adjacent free vertices in $N$. But then replacing $e$ with $(x,y)$ in $N$ produces a matching with $t$ free and larger intersection with $M$. This is a contradiction. $\square$
Now we can prove your result:
In every bipartite graph $G$, there exists a vertex $v$ that is matched in every maximum matching.
Proof: Suppose otherwise. Then every vertex in $G$ must be missed by a maximum matching. Working with the components of $G$, we may assume without loss of generality that it is connected. The previous theorem implies that the graph is factor critical. In particular, this means that the graph's two components, $A$ and $B$, have sizes $n$ and $n+1$ respectively (with maximum matching of size $n$). But then a maximum matching cannot possibly miss a vertex in $A$ for it must then miss two vertices in $B$. This contradicts the fact that the graph is factor critical. $\square$.
In passing, I mention that this result almost immediately implies König's theorem. Let $G$ be a minimal counter-example. Let $v$ be a vertex which is matched by every maximal matching of $G$. Then we must have $\nu(G\setminus v) < \nu(G)$ for otherwise there exists a maximal matching of $G\setminus v$ of size $\nu(G)$, i.e. a maximal matching of $G$ which misses $v$. By minimality of $G$, it follows that $\tau(G\setminus v) = \nu(G\setminus v) < \nu(G)$. Adding back $v$ to a minimal cover of $G\setminus v$ then gives $\tau(G) \le \nu(G)$ as required.
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2019-11-13 22:55:47
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https://math.stackexchange.com/questions/3555018/approaches-for-the-study-of-rings
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# Approaches for the study of rings
When, as a student in my first weeks, I first read the definition of a group, the concept of groups looked very natural and motivated to me. But when I read the definition of a ring some time later, I had no idea what rings should be about and how they were motivated. Especially the distributive property let me wonder how much of the multiplicative structure is already determined by the addition.
All of this is has been two years ago. When I recently started to read Algebra: Chapter $$\it 0$$ by Paolo Aluffi, a lot of thinks I had not understood before, became clear to me. Mostly because of the use of categorical language which really works well with the way I think about mathematics.
I even came to peace with the concept of rings, when Aluffi explained them as coming up in the context of studying the homsets of abelian groups (we take the group operation as addition and the composition as multiplication).
I have been very happy about this explanation, but then, instead of studying ring by using this approach, Aluffi used the 'classical' definition of rings again.
My question is: Why?
Is it not much more promising to study rings by studying the homsets of abelian groups? I know that there is not any loose of information by just using the classical definition, but using the definition via homsets looks a lot more natural to me.
I know some category theory (at least I read Emily Rhiels book Category Theory in Context) so I could read I bit trough nLab and soon found out that a ring is equivalent to a preadditive category with only one object (encapturing pretty much what I called 'homset definiton'). So why don't we study thous preadditive categories instead? They seem to have far more 'visible structure' then rings have in their usual definition.
(I know that this is kind of a soft question. I do not look in particular for a kind of 'rigorous answer', but rather for your personal view towards this.)
• endomorphisms of an Abelian group are an example of a ring – J. W. Tanner Feb 21 '20 at 13:21
• The study of rings appears in a first course in abstract algebra. I don't see why we should start with "preadditive categories instead". The basic examples are $\Bbb Z$, or other number rings like $\Bbb Z[i]$, matrix rings, polynomial rings, endomorphism rings etc. – Dietrich Burde Feb 21 '20 at 13:40
• What is the definition that you propose is better than the classical one? The classical one makes all the basic properties very transparent. Not to mention, it would add a lot of overhead learning more advanced concepts. I don't know what definition you're using, exactly, but it seems to me it would probably bury the basic properties. Anyhow, if you have at least one appealing viewpoint on rings, it is unclear to me why the existence of "the classical viewpoint" would sour you on the subject. – rschwieb Feb 21 '20 at 13:44
• I suspect you'll enjoy reading about category-theoretic generalizations of these "Cayley" representations (beyond what you mention for rings), e.g. see the book by Pultr and Trnková mentioned in this answer – Bill Dubuque Feb 21 '20 at 15:11
• Every group is a subgroup of a symmetric group - this doesn't mean that we should only study symmetric groups. Similarly, every ring is a subring of the endomorphism ring of an abelian group, but there is good reason to not restrict ourselves to this case. – HallaSurvivor Feb 21 '20 at 19:21
I wanted to expand on my comment, but my expansion was getting too big, so I've decided to add it as an answer. Just like the question, my answer is somewhat soft (and is certainly opinionated) so take it with a grain of salt.
Every group is a subgroup of a symmetric group - this doesn't mean that we should only study symmetric groups. Similarly, every ring is a subring of the endomorphism ring of an abelian group, but there is good reason to not restrict ourselves to this case.
Similarly, wrt the "categorical" definitions. Aluffi presents (somewhat cheekily)
Joke 1.1: A group is a groupoid with a single object.
On the very next page, Aluffi gives "Definition 1.2" which is the traditional definition of a group. I'm sure he could have included a similar joke regarding the definition of rings (instead of giving homsets as an example of rings), but doubtless he would have also settled on the "traditional" definition of rings as his real definition.
The categorical definitions are nice in that they let us see relations between structures, and often give us tools for proving something in multiple areas simultaneously by abstracting almost everything specific away, leaving only the structure of the problem.
This can be useful, but only after having seen the "traditional" definitions of our objects. First and foremost, the traditional definitions require no background knowledge. When we give the definition of a ring, our examples can be things like $$\mathbb{Z}$$ and $$\mathbb{Q}[x]$$. Extremely concrete objects that we have been playing with since middle school. When we give the definition of a ring in terms of homsets of abelian groups, suddenly we don't have our concrete examples anymore, or at least, we don't obviously have them. This is pedagogically worse, as it obscures why rings are the way they are -- because they generalize things we already care about.
Not to mention the categorical definitions often presuppose you know some category theory! To say that "a group is a groupoid with one element" is nice and all, but it's only helpful if your audience understands groupoids! When introducing a new topic, we want to draw analogies between the new topic and things the audience already knows. Since categorical tools are at the top of the abstraction hierarchy, the intuition we get for, say, homsets, comes from our knowledge of rings. Not the other way around. This is because rings are "closer" to simple things like $$\mathbb{Z}$$ than homsets are.
I agree that rings are, at face value, the grossest of the "big three" algebraic structures (groups/rings/fields). But through the study of their modules, and eventually through algebraic geometry, I learned to love them (though noncommutative rings still scare me...).
The moral is that the traditional definition is traditional for a reason, and trying to look for abstraction too soon is likely to confuse rather than enlighten. To learn to love rings, you just need to spend some more time with them, on their own terms. They arise very naturally in algebraic geometry -- perhaps that is a good place to look for your justification. At the very least, I hope you can see that, even if the categorical definition is better for you (which I'm still not convinced it is), the given definition is likely to be more useful for more people.
I hope this helps ^_^
We have the four basic operations of numbers: $$+,\, -,\,\cdot,\,/$$.
Rings (with $$+,\, -, \, \cdot$$) are able to do number theory in an abstract level by talking about divisibility, factorization. E.g. according to Dedekind, ideals represent 'ideal divisors' which in general might not be represented by an element of the ring.
If we drop commutativity (or drop $$+,-, 0$$ from the definition of a skew field), we receive groups.
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2021-06-24 08:07:45
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https://www.physicsforums.com/threads/class-example-limit-of-a-function-using-definition.550205/
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# Homework Help: Class example: limit of a function using definition
1. Nov 13, 2011
### PirateFan308
I am having trouble understanding how to find the limit of a function (using the definition of a limit). I have a class example, and was wondering if anyone could walk me through the steps.
1. The problem statement, all variables and given/known data
Using the definition of the limit to show that limx→2(x2)=4
f(x) = x2
c=2
L=4
Given an arbitrary ε>0, take δ=min{1,ε/5}
If x≠2 and |x-2|<δ then |x-2|<1 and |x-2|< ε/5
|f(x)-L| = |x2-4| = |(x-2)(x+2)| = |x-2||x+2|
|x-2|<1 => 1<x<3 => 3<x+2<5 => |x+2|<5
|x-2||x+2| < (ε/5)(5) = ε so |f(x)-L|<ε
2. Relevant equations
We say that lim f(x)x→c=L if:
$\forall$ε>0 $\exists$δ>0 $\forall$x$\in$dom f if x≠c and |x-c|<δ then |f(x)-ε|<L
3. The attempt at a solution
The biggest thing I am confused about is how the professor got δ? Did he have to do the later work first and then went back and plugged in the answer he got?
Also, in the definition, it says that then |f(x)-ε|<L but we ended up getting |f(x)-L|<ε. Why is this? I understand that we can rearrange the equation, but then doesn't this mess up the absolute value signs?
2. Nov 13, 2011
### SammyS
Staff Emeritus
"The biggest thing I am confused about is how the professor got δ? Did he have to do the later work first and then went back and plugged in the answer he got?"
Your professor likely did some scratch work, starting with |x2-4|<ε, and then getting his result for δ.
"in the definition, it says that then |f(x)-ε|<L but we ended up getting |f(x)-L|<ε"
It should be |f(x)-L|<ε in the definition.
3. Nov 13, 2011
### Dick
|f(x)-epsilon|<L is a typo. |f(x)-L|<epsilon is the correct form. And yes, the professor figured out a delta using the later work and then went back and plugged it in.
4. Nov 13, 2011
### PirateFan308
Another question, is there more than one δ that will prove this?
Say, Given an arbitrary ε>0, take δ=min{2,ε/6}
If x≠2 and |x-2|<δ, then |x-2|<2 and |x-2|<ε/6
|f(x)-L| = |x2-4| = |(x+2)(x-2)| = |x+2||x-2|
|x-2|<2 => -2<x-2<2 => 0<x<4 => 2<x+2<6 => |x+2|<6
|x-2||x+2| < (6)(ε/6) = ε so |f(x)-L|<ε
5. Nov 13, 2011
### Dick
Sure, that choice works just as well.
6. Nov 13, 2011
### SammyS
Staff Emeritus
Yes, there are many ways to come up with δ .
7. Nov 13, 2011
### PirateFan308
Why is it that I must say d=min{1,ε/5}? Would it also work if I said that δ=1,ε/5. I'm a bit fuzzy on how the "min" makes this true, or the absence of "min" makes it false.
8. Nov 13, 2011
### Harrisonized
Last edited: Nov 13, 2011
9. Nov 13, 2011
### SammyS
Staff Emeritus
If ε > 5, then if you say that δ > ε/5, the proof won't work.
Let's say ε = 10.
Then the claim would be that δ = 2 will satisfy the definition.
But if x=3.9, then f(3.99)=15.21, so |f(3.99)-2| = 13.21 > 10
Last edited: Nov 13, 2011
10. Nov 13, 2011
### PirateFan308
So is it standard procedure to always take δ=min if there is more than one condition? Will it ever be wrong for me to make δ=min ?
11. Nov 13, 2011
### Dick
In the proof you used that d<=1 AND d<=epsilon/5. min(1,epsilon/5) is less than or equal to both of them. d=1 doesn't work if you pick a small epsilon. d=epsilon/5 doesn't work if you pick a large epsilon. Try it.
12. Nov 14, 2011
### PirateFan308
Thank you! This makes so much more sense now!
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2018-11-19 01:11:59
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http://math.stackexchange.com/questions/206966/projective-spaces-disjoint-union
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# Projective spaces - Disjoint union
I was trying to prove that $\mathbb{RP^{3}}$ is homeomorphic to que quotient space of the disjoint union $\mathbb{RP^{2}}$$\bigsqcup$$B$ by the "gluing" $f$: $\partial$$B \rightarrow \mathbb{RP^{2}}, where B is the closed unit ball in \mathbb{R^{3}}. My idea is to construct a quotient map from \mathbb{S^{3}} to the quotient space of \mathbb{RP^{2}}$$\bigsqcup$$B$ which makes the same identifications that the quotient map which constructs $\mathbb{RP^{3}}$, and conclude by a widely-known theorem that these two spaces are, in fact, homeomorphic. I was trying to do it by composition of some continuous functions, using the closed map lemma to help me, but I couldn't go any further. Do you have any hint or suggestion?
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Do you know CW-complexes? – Chris Gerig Oct 4 '12 at 5:29
No, I haven't studied it yet... – Br09 Oct 4 '12 at 16:02
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2014-09-22 06:44:09
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https://math.stackexchange.com/questions/3292255/semigroup-structure-of-principal-ideals-under-products-in-a-dedekind-domain
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# Semigroup structure of principal ideals (under products) in a Dedekind domain
Specifically, let $$R$$ be our Dedakind domain, I am trying to show that the structure of the principal ideals is that of a free abelian semigroup (isomorphic to $$\mathbb{N}\times\mathbb{N}\times\cdots$$) $$\iff$$ $$R$$ is a PID.
Now I know in this case PID $$\iff$$ UFD. So I suppose that having ideal $$(a) = (x_1)(x_2) = (y_1)(y_2)$$ where $$x_i$$ and $$y_i$$ are irreducible and $$x_i\neq y_j$$ must come into play. But I can't really see how to make this relevant as this reasoning would also apply to the group of fractional principal ideals, which must be free abelian since it's a subgroup of a the free abelian group of ideals (freeness coming from unique factorization into primes). I can't even see in the abstract how a sub-semigroup (subset closed under the associative operation) of a free abelian semi-group could not also be free abelian.
Any guidance would be much appreciated. By the way I am trying to work through Number Fields by Marcus and this is part of problem 31 from chapter 3, just in case that helps with answering. Thanks.
After posting I think I understand. Funny how that works. Anyway first we must be more precise about what is a free abelian semigroup: including $$0\in\mathbb{N}$$ a free abelian semigroup is something isomorphic to $$\mathbb{N}\times\mathbb{N}\times\cdots\backslash \{(0,0,0,\cdots)\}$$.
Anyway, going back to the non-unique factorization, let $$a=x_1x_2=y_1y_2$$ be two different factorizations into irreducible elements so we have $$(a)=(x_1)(x_2)=(y_1)(y_2)$$. Since the $$x_i$$ (and $$y_i$$) are irreducible, than as elements of our semigroup they cannot be written as a product of other elements. So if we have a free abelian semigroup on our hands, the elements $$(x_1)$$, $$(x_2)$$, $$(y_1)$$ and $$(y_2)$$ must be "basis" elements. That is, they must map to elements like $$(1,0,0,0,0,\cdots)$$, $$(0,1,0,0,0,\cdots)$$ etc. This is not the case in a free abelian group since things can always be written as the product of other elements (e.g. $$(x_1) = (a)(x_2)^{-1}$$ in the group of principal fractional ideals) which is what was tripping me up before I think.
Anyway, with $$(x_1)$$, $$(x_2)$$, $$(y_1)$$ and $$(y_2)$$ as "basis" elements we have a contradiction since $$(a)$$ has multiple descriptions in terms of basis elements which never could happen in a free abelian semi-group: we have the relation $$(x_1)(x_2)=(y_1)(y_2)$$.
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2020-01-27 00:19:14
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http://www.longluo.me/blog/2022/04/12/Leetcode-merge-two-binary-trees/
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By Long Luo
Here are 4 approaches to solve this problem in Java: Recursion, Iteration, BFS and DFS.
# Recursion
## Method 1: New Tree
We can create a new Tree, each $\texttt{TreeNode}$ value is sum of two nodes.
## Analysis
• Time Complexity: $O(min(m, n))$
• Space Complexity: $O(min(m, n))$
## Method 2
Traverse both the given trees in a PreOrder style.
At every step, check if the current node exists for both the trees. If one of these children happens to be null, we return the child of the other tree to be added as a child subtree to the calling parent node in the first tree.
We can add the values in the current nodes of both the trees and update the value in the current node of the first tree to reflect this sum obtained.
Then we call the $\texttt{mergeTrees()}$ with the left children and then with the right children of the current nodes of the two trees.
At the end, the first tree will represent the required resultant merged binary tree.
## Analysis
• Time Complexity: $O(min(m, n))$
• Space Complexity: $O(min(m, n))$
# Iteration
We can also traverse the two trees by make use of a stack to do so.
Each entry in the Stack strores data in the form $[\textit{node}_{tree1}, \textit{node}_{tree2}]$.
1. We push the root nodes of both the trees onto the stack.
2. At every step, we remove a node pair from the top of the stack.
3. For every node pair removed, we add the values corresponding to the two nodes and update the value of the corresponding node in the first tree.
4. If root1.left != null && root2.left != null, we push the left child(pair) of both the trees onto the stack.
5. If $root1.left == null$, we append the left child(subtree) of the second tree to the current node of the first tree. We do the same for the right child pair as well.
6. If, at any step, both the current nodes are null, we continue with popping the next nodes from the stack.
## Analysis
• Time Complexity: $O(min(m, n))$
• Space Complexity: $O(min(m, n))$
# BFS
BFS is like the Iteration method, it create a new Tree.
The BFS code is not neat, I have refactor it.
## Analysis
• Time Complexity: $O(min(m, n))$
• Space Complexity: $O(min(m, n))$
# DFS
DFS is the same as the recursion method, just a little bit difference.
## Analysis
• Time Complexity: $O(min(m, n))$
• Space Complexity: $O(min(m, n))$
All suggestions are welcome.
If you have any query or suggestion please comment below.
Please upvote👍 if you like💗 it. Thank you:-)
Explore More Leetcode Solutions. 😉😃💗
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2023-03-26 11:48:58
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http://en.wikipedia.org/wiki/Sierpinski_gasket
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# Sierpinski triangle
Sierpinski triangle
Generated using a random algorithm
Sierpinski triangle in logic: The first 16 conjunctions of lexicographically ordered arguments
The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence A001317 in OEIS)
The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries prior to the work of Sierpiński.
## Constructions
There are many different ways of constructing the Sierpinski triangle.
### Removing triangles
The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
2. Subdivide it into four smaller congruent equilateral triangles and remove the central one.
3. Repeat step 2 with each of the remaining smaller triangles
Each removed triangle (a trema) is topologically an open set.[1] This process of recursively removing triangles is an example of a finite subdivision rule.
### Shrinking and duplication
The same sequence of shapes, converging to the Sierpinski triangle, can alternatively be generated by the following steps:
1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
3. Repeat step 2 with each of the smaller triangles (image 3 and so on).
Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."[2]
The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let $d_a$ note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation $d_a$ U $d_b$ U $d_c$.
This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.
### Chaos game
If one takes a point and applies each of the transformations $d_a$, $d_b$, and $d_c$ to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:
Start by labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set vn+1 = ½ ( vn + prn ), where rn is a random number 1, 2 or 3. Draw the points v1 to v. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpinski triangle. If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle. If v1 is outside the triangle, the only way vn will land on the actual triangle, is if vn is on what would be part of the triangle, if the triangle was infinitely large.
Animated creation of a Sierpinski triangle using the chaos game
Animated construction of a Sierpinski triangle
Or more simply:
1. Take 3 points in a plane to form a triangle, you need not draw it.
2. Randomly select any point inside the triangle and consider that your current position.
3. Randomly select any one of the 3 vertex points.
4. Move half the distance from your current position to the selected vertex.
5. Plot the current position.
6. Repeat from step 3.
Note: This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points. It is interesting to do this with pencil and paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
Sierpinski triangle using an iterated function system
Construction of the Sierpiński arrowhead curve
Another construction for the Sierpinski triangle shows that it can be constructed as a curve in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake:
2. Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with it.
The resulting fractal curve is called the Sierpiński arrowhead curve, and its limiting shape is the Sierpinski triangle.[3]
### Cellular automata
The Sierpinski triangle also appears in certain cellular automata (such as Rule 90), including those relating to Conway's Game of Life. For instance, the life-like cellular automaton automaton B1/S12 when applied to a single cell will generate four approximations of the Sierpinski triangle.[4] The time-space diagram of a replicator pattern in a cellular automaton also often resembles a Sierpinski triangle.[5]
### Pascal's triangle
If one takes Pascal's triangle with 2n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpinski triangle.[6]
### Towers of Hanoi
The Towers of Hanoi puzzle involves moving disks of different sizes between three pegs, maintaining the property that no disk is ever placed on top of a smaller disk. The states of an n-disk puzzle, and the allowable moves from one state to another, form an undirected graph that can be represented geometrically as the intersection graph of the set of triangles remaining after the nth step in the construction of the Sierpinski triangle. Thus, in the limit as n goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle.[7]
## Properties
For integer number of dimensions d, when doubling a side of an object, 2 d copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For Sierpinski triangle doubling its side creates 3 copies of itself. Thus Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from solving 2 d = 3 for d.[8]
The area of a Sierpinski triangle is zero (in Lebesgue measure). The area remaining after each iteration is clearly 3/4 of the area from the previous iteration, and an infinite number of iterations results in zero.[9]
The points of a Sierpinski triangle have a simple characterization in Barycentric coordinates.[10] If a point has coordinates (0.u1u2u3…,0.v1v2v3…,0.w1w2w3…), expressed as Binary numbers, then the point is in Sierpinski's triangle if and only if ui+vi+wi=1 for all i.
## Analogues in higher dimensions
A Sierpinski square-based pyramid and its 'inverse'
A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity in this 2-dimensional projected view, so that the resulting triangle could be a 2D fractal in itself.
The tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. This can also be done with a square pyramid and five copies instead. A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration.
The initial surface area of the (iteration-0) tetrahedron of side-length L is $L^2 \sqrt{3}$. At the next iteration, the side-length is halved
$L \rightarrow { L \over 2 }$
and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:
$4 \left( \left( {L \over 2} \right)^2 \sqrt{3} \right) = 4 { {L^2} \over 4 } \sqrt{3} = L^2 \sqrt{3}.$
This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of the tetrahedron in the previous iteration, there are 4 times as many—thus maintaining a constant total surface area.
The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptotically approaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has no 3-dimensional character. The Hausdorff dimension of such a construction is $\textstyle\frac{\ln 4}{\ln 2}=2$ which agrees with the finite area of the figure. (A Hausdorff dimension strictly between 2 and 3 would indicate 0 volume and infinite area.)
## History
Wacław Sierpiński described the Sierpinski triangle in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy,[11] and other places of central Italy, for carpets in many places such as the nave of the Roman Basilica of Santa Maria in Cosmedin,[12] and for isolated triangles positioned in rotae in several churches and Basiliche.[13] In the case of the isolated triangle, it is interesting to notice that the iteration is at least of three levels.
## References
1. ^ "Sierpinski Gasket by Trema Removal"
2. ^ Michael Barnsley, et al."V-variable fractals and superfractals" PDF (2.22 MB)
3. ^ Prusinkiewicz, P. (1986), "Graphical applications of L−systems", Proceedings of Graphics Interface '86 / Vision Interface '86, pp. 247–253.
4. ^ Rumpf, Thomas (2010), "Conway's Game of Life accelerated with OpenCL", Proceedings of the Eleventh International Conference on Membrane Computing (CMC 11), pp. 459–462.
5. ^ Bilotta, Eleonora; Pantano, Pietro (Summer 2005), "Emergent patterning phenomena in 2D cellular automata", Artificial Life 11 (3): 339–362, doi:10.1162/1064546054407167.
6. ^ Stewart, Ian (2006), How to Cut a Cake: And other mathematical conundrums, Oxford University Press, p. 145, ISBN 9780191500718.
7. ^ Romik, Dan (2006), "Shortest paths in the Tower of Hanoi graph and finite automata", SIAM Journal on Discrete Mathematics 20 (3): 610–622 (electronic), arXiv:math.CO/0310109, doi:10.1137/050628660, MR 2272218.
8. ^ Falconer, Kenneth (1990). Fractal geometry: mathematical foundations and applications. Chichester: John Wiley. p. 120. ISBN 0-471-92287-0. Zbl 0689.28003.
9. ^ Helmberg, Gilbert (2007), Getting Acquainted with Fractals, Walter de Gruyter, p. 41, ISBN 9783110190922.
10. ^ http://www.cut-the-knot.org/ctk/Sierpinski.shtml
11. ^ Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, pp. 43, 873
12. ^
13. ^ Tedeschini Lalli, Elisa (2011), Sierpinski Triangles in Stone on Medieval Floors in Rome, APLIMAT Journal of Applied Mathematics n.4, pp. 114, 122
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2014-08-31 07:27:53
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https://www.nature.com/articles/s41598-017-11266-1?error=cookies_not_supported&code=a38da3d5-949c-4fab-b4b0-f0dbbb71735c
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## Introduction
Constructing a physical computing machine, whether a classical or a quantum one, requires, inescapably, the implementation of an error correcting mechanism that guards against the noise picked up from the environment and the imperfections in the operations being performed1,2,3. Early in the development of both classical and quantum computers, “threshold” theorems were proven to show the existence of encoding schemes which reliably store and process information in a “logical” set of bits (or qubits), by encoding it redundantly on top of a bigger set of less reliable “physical” bits (or qubits), as long as the error rate on the physical layer is smaller than a fixed threshold4, 5. The vast majority of quantum error correcting codes fall in the class of stabilizer codes (a generalization of the classical linear codes)6. They are characterized by the group of stabilizer operators that preserve the logical states (similarly to the list of constraints represented by the parity check matrix H for classical linear codes). The list of nontrivial stabilizer operator measurements (or violated parity constraints for a classical code) is called the syndrome of the error. While providing for efficient encoding, linear and stabilizer codes do not necessarily have known efficient decoding algorithms that can deduce from a given syndrome what errors have occurred.
In the general case decoding a stabilizer code is an NP-hard problem. An active area of research is the design of codes with some additional algebraic structure that permits efficient decoders, but still retains high rates (ratio of logical to physical qubits) with acceptable distances (maximal number of correctable errors on the physical qubits). Schemes like the CSS approach7,8,9 permit the creation of quantum codes from classical codes, but they do not guarantee that the decoder that worked for the classical code still works for the quantum one. A particularly interesting example is the class of LDPC codes10, 11 which are high-performing classical codes with efficient decoders, however those decoders do not work for the quantum LDPC codes12.
Here we present a decoding algorithm that can be applied to any stabilizer code — the decoder employs machine learning techniques to “learn” any structures that would make the approximate decoding problem easier than the general NP-hard decoding problem: it “learns” the probability distributions of errors conditioned on a given syndrome and efficiently uses samples from that distribution in order to predict probable errors. The conditional probability distribution is encoded in a deep neural network. The “learning” involves training the neural network on pairs of errors and corresponding syndromes (generated from an error model for the physical qubits and a parity check matrix for the code in use). We test the algorithm on the toric code (Fig. 1a) definied on a two-dimensional lattice on a torus13. Since the toric code has low-weight local stabilizers, it is also a quantum LDPC code with structure that impedes typical belief propagation algorithms. Our decoder significantly outperforms the standard “minimal-weight perfect matching” (MWPM) decoder14, 15. Moreover, it has comparable threshold with the best renormalization group decoders16. For code-sizes up to 200 physical qubits the decoder is practical and we discuss how to extend our neural network architecture to negate the inefficiencies that kick in at that stage.
Machine learning techniques, specifically neural networks, have been gaining popularity over the last year, in particular with the recent developments in using restricted Boltzmann machines for describing the ground state of many-body systems17 or convolutional networks for identifying phases of matter18. A preprint on the use of restricted Boltzmann machines to decoding the toric code has been available for a few months as well19, however that architecture does not yet outperform known decoders like MWPM and has been tested only on the Z syndrome on lattices no bigger than 5-by-5. At the time of submission of this manuscript two other related preprints were made available: a fast neural network decoder for small surface codes, that however also does not significantly outperform MWPM20, and a recurrent neural network decoder outperforming MWPM as evaluated on a 17 qubit surface code21. It is worth noting as well that over the last few months work has started on deep learning methods for decoding classical algebraic codes22.
## Results
For testing purposes we trained our neural decoder (depicted in Fig. 1b) on the toric code, which already has a number of known decoding algorithms specifically tuned for its lattice structure. The evaluation was done under the depolarization error model. Our algorithm significantly outperforms the standard MWPM decoder. The comparison of the two decoders in Fig. 2 shows a threshold single-qubit error which is nearly 2 percentage points higher for the new algorithm (around 16.4% for the depolarization error model), and the fraction of correctly decoded errors is consistently around 10 percentage points higher than the fraction of errors corrected by MWPM. Furthermore, the neural decoder threshold compares favorably to renormalization group decoders16 (threshold of 15.2%), and decoders explicitly tuned to correlations between Z and X errors23 (threshold of 13.3% for a triangular lattice, as it is tuned for asymmetric codes). To our knowledge only a renormalization group decoder24 enhanced by a sparse code decoder12 reaches a similar threshold (16.4%). It is worth noting that the sampling procedure in our decoder makes it impractically slow for codes of more than 200 physical qubits, while other decoders remain practical. On the other hand, the neural architecture is versatile enough to be applied to any stabilizer code, unlike the other decoders discussed here, which are limited to only topological codes. The best of both worlds — record threshold and fast decoding — should be achievable if we couple the renormalization decoder of24 with our neural decoder (instead of the currently suggested sparse code decoder12), however this will be applicable only to topological codes. We discuss other ways to avoid the inefficiencies in our decoder without compromising its ability to “learn” to decode any stabilizer code.
After being properly trained for a given error rate of a particular error model, the neural network at the heart of our decoder becomes a compact approximate representation of the probability distribution of errors that can occur. The decoding algorithm consist of inputing the measured syndrome in the neural network, interpreting the output as a probability distribution of the errors conditioned on the given syndrome, and repeatedly sampling from that distribution. The performance of the decoder scales monotonically with the size of the network, up to a point of diminishing returns where using more than about 15 hidden layers (for a distance 5 code) stops providing improvements.
The significant gain in the threshold value relative to some known decoders can be traced to two characteristics of the neural network (discussed in more details in the Methods section). Firstly, the neural network is trained on (stabilizer, error) pairs generated from the error model, therefore it is optimized directly for producing “most probable error”, not for finding an imperfect proxy like “error with lowest energy” as is the case for MWPM. Secondly (depicted in Fig. 3), it learns the Z and X stabilizers together, hence it can encode correlations between them in its structure. Namely, in a typical depolarization error models, one third of the errors are Y errors (equivalent to both X and Z error happening), therefore the knowledge of this correlation can be a useful resource for decoding. Other decoders need significant modifications to even partially employ those correlations in decoding23.
## Methods
Neural networks are particularly efficient tools for function approximation25, where a function f:x→f(x) is to be learned from large amount of training data given in the form of pairs (x,f(x)). The input x is set as the value of the input layer of neurons. Each of those neurons is connected through axons with each neuron of the next layer (the first “hidden” layer). Multiple hidden layers of neurons can be connected together in this fashion in order to construct a deeper neural network. The last layer of the network is the output layer - its value represents f learned (x). The value of a neuron (i.e. its activation value) is calculated as a weighted sum of the activation values of the neurons connected to it from the previous layer. That sum is then passed through a non-linear function (called the activation function). This activation value is then further passed on to the neurons of the next layer, where the process is repeated until it reaches the output layer. The weights in the sums (i.e. the strength of connections between the neurons) are parameters which are optimized through stochastic gradient descent in order to minimize the distance between f learned and f calculated on the training data. The choice of activation function, the size of the hidden layers, and the step size for gradient descent (also called the hyperparameters) are decided in advance, before training. Current best practices include performing a random search to find the best hyperparameters.
In the particular case of decoding a stabilizer quantum error correcting code we want to map syndromes to corresponding physical errors, hence, we take the input layer to be the syndrome (obtained from measuring the stabilizers). For instance, for a toric code of lattice size 9-by-9 we have to measure 81 plaquette operators and 81 star operators for a total of 162 input neurons (having value 0 if the syndrome is trivial and 1 if not). Similarly, we set the output layer to be the prediction for what physical errors occurred (typically represented in the Heisenberg picture, thanks to the Gottesman– Knill theorem). Using the same example, we have 162 physical qubits and we need to track their eigenvalues under both Z and X operators, requiring a total of 324 output neurons (having value 0 if no error has occurred and value 1 otherwise).
To completely define the neural network architecture we set the activation functions of the hidden layers to tanh and the activation of the output layer to the sigmoid function σ(x) = (1 − e x)−1[0, 1]. The size of the hidden layer was set to four times the size of the input layer. These decisions were reached after an exhaustive search over possible hyperparameters tested on toric codes of distance 3 to 6, and proved to work well for bigger codes as well. The number of hidden layers was varied - deeper networks produce better approximations up to a point of diminishing returns around 15 layers. The step size for the gradient descent (a.k.a. the learning rate) was annealed - gradually lowered, in order to permit rapidly reaching the minimum. The distance measure between training and evaluation data that is being minimized by the gradient descent is their binary crossentropy (a measure of difference between two probability distributions discussed below).
The training was done over one billion (syndrome, error) pairs in batches of 512, taking about a day of GPU wall time for a 5-by-5 toric code. The pairs were generating on the fly, by first generating a sample error from the given error model (this training set can also be efficiently generated directly on the experimental hardware), and then obtaining the corresponding syndrome by a dot product with the parity check matrix. The error model used for each physical qubit was qubit depolarization, parametrized by qubit fidelity p - the probability of no error happening on a given qubit, or equivalently depolarization rate 1 − p. Under this model, Z, X, and Y (consecutive Z and X) errors had equal probabilities of 1/3(1 − p). For each value of p we trained a new network, however the results showed some robustness to testing a neural network at an error rate different from the one at which it was trained.
The performance of the network was improved if we normalize the input values to have an average of 0 and a standard deviation of 1. For a depolarization error rate 1 − p, the rate at which a Z eigenvalue flips is P e = 2/3(1 − p) and independently the rate for X flips is the same. In the example of the toric code the rate of non-trivial stabilizer measurements will be the same for Z and for X, namely P s = 4q 3(1 − q) + 4q(1 − q)3 and the variance will be V s = P s − P s 2.
At this point we have not discussed yet how to use the fully trained neural network in decoding. A trained network can efficiently evaluate the approximation of the decoding function (from here on referred to as DECODE: syndrome → error), so all Alice needs to do in order to perform error correction on her quantum memory is to measure the syndrome and run the neural network forward to evaluate DECODE(syndrome). However, the neural network is a continuous function and an imperfect approximation, therefore the values in DECODE(syndrome) will not be discrete zeros and ones, rather they will be real numbers between zero and one. A common way to use and interpret those values is to view them as a probability distribution over possible errors, i.e. the i-th value in the array DECODE(syndrome) is a real number between zero and one equal to the probability of the i-th qubit experiencing a flip (half of the array corresponds to Z errors and half of the array corresponds to X errors). This interpretation is reinforced by our use of binary crossentropy as an optimization target during training. In order to deduce what error has occurred we sample this probability distribution. We verify the correctness of the sample by computing the syndrome that the predicted error would cause - if it differs from the given syndrome we resample. This sampling procedure is present in ref. 19 as well, however we further employ a simple “hard decision belief propagation/message passing” sampling, which can speed up the sampling process by an order of magnitude: we resample only the qubits taking part in the stabilizer measurement corresponding to the incorrect elements of the syndrome (Fig. 4).
### Data availability
The code for building, training, and evaluating the neural network decoder is publicly available on the authors’ web page, and shell scripts with the parameters for the presented figures are available upon request. Pretrained neural networks can be provided as well.
## Discussion
On first sight our decoder implementation can look like a look-up table implementation, however we would like to stress the immense compression of data that the neural network achieves. Firstly, one can consider the size of the neural network itself. For a code on N physical qubits the number of parameters needed to describe a neural decoder of L layers will be $${\mathscr{O}}({N}^{2}L)$$ or on the order of thousands for the codes we tested. Moreover, the size of the training dataset for the codes we tested did not exceed 10 billion, and it can be made orders of magnitude smaller if we reuse samples in the stochastic gradient descent (a common approach taken in training). On the other hand, the size of a complete lookup table would be on the order of $${\mathscr{O}}({4}^{N})$$. Even if we take only the most probable errors (and discard the errors that have less than 5% chance of occurring), at depolarization rate of 0.1 we need a lookup table bigger than 1012 for a distance 5 toric code (50 qubits), bigger than 1023 for distance 7 toric code (98 qubits), and bigger than 1037 for distance 9 toric code (162 qubits).
Thanks to this compression, to the direct optimization for most probable error, and to the ease of including knowledge of error correlations in the decoding procedure, the algorithm presented here is one of the best choices for decoding stabilizer codes of less than 200 qubits. While we used the toric code for our testing, there is nothing in our design that has knowledge of the specific structure of that code - the neural decoder can be applied to the decoding of any stabilizer code.
Due to the probabilistic nature of the sampling, the decoder becomes impractically inefficient for codes bigger than roughly 200 qubits as one can see in Fig. 5. This can be attributed to two characteristics of our algorithm: we use a simple hard-decision message passing algorithm in our sampling instead of a more advanced belief propagation algorithm seeded by output of the neural network; additionally, our neural network learns only the marginal probabilities for errors on each qubit, without providing the correlations between those errors. A more advanced neural network could address this problem by providing correlation information in its output layer. Our focus forward goes beyond that: we can consider recurrent generative networks26 that have the belief propagation as part of their recurrent structure.
While this decoder is general and it can be applied to any stabilizer code, one can also design neural network architectures that specifically exploit the lattice structure and translational symmetry of the toric code. For instance, convolutional neural networks are well adapted for processing 2D data. Moreover thanks to the translational symmetry one can envision a decoder that is trained on a fixed patch of the code and it can be used for toric codes of any size. As already mentioned, our decoder can readily replace the sparse code decoder12 used as part of the renormalization group decoder of 24, hence providing great decoding speed and high threshold values.
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2023-01-29 07:07:25
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http://physicscatalyst.com/maths/sets1.php
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# Sets
## Power Set
• The collection of all subsets of a set $X$ is called the power set of $X$. It is denoted by $P(X)$. In $P(X)$, every element is a set.
• if $X = \left\{ {1,2,3} \right\}$, then
$X\left( A \right) = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {1,3} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\}} \right\}$
Also, note that $n\left[ {X\left( A \right)} \right] = 8 = {2^3}$
• In general, if $X$ is a set with $n\left( X \right) = m$, then it can be shown that
$n\left[ {P\left( A \right)} \right] = {2^m}$
## Universal Set
A Universal is the set of all elements under consideration, denoted by capital U.
## Venn diagram
Venn diagrams were introduced in 1880 by John Venn (1834–1923). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles. Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while another circle may represent the set of all tables
## Operation on Sets
### Union of Sets
The union of two sets $A$ and $B$ is the set $C$ which consists of all those elements which are either in $A$ or in $B$ (including those which are in both). In symbols, we write.
$A \cup B = \left\{ {x:x \in A{\rm{ or }}x \in B} \right\}$
#### Some Properties of the Operation of Union
• Commutative law : $X \cup Y = Y \cup X$
• Associative law : $\left( {X \cup Y} \right) \cup Z = X \cup \left( {Y \cup Z} \right)$
• Law of identity element, $\phi$ is the identity of $\cup$ : $X \cup \phi = X$
• Idempotent law : $X \cup X = X$
• Law of U : $U \cup X = U$
### Intersection of Sets
The Intersection of two sets $A$ and $B$ is the set $C$ which consists of all those elements which are present in both $A$ and $B$ . In symbols, we write.
$A \cap B = \left\{ {x:x \in A{\rm{ and }}x \in B} \right\}$
#### Some Properties of Operation of Intersection
• Commutative law : $X \cap Y = Y \cap X$
• Associative law : $\left( {X \cap Y} \right) \cap Z = X \cap \left( {Y \cap Z} \right)$
• Law of $\cap$ and $U$ : $\phi \cap X = \phi$ , $U \cap X = X$
• Idempotent law : $X \cap X = X$
• Distributive law : $X \cap \left( {Y \cup Z} \right) = \left( {X \cap Y} \right) \cup \left( {X \cap Z} \right)$
### Difference of set
The difference of two sets $A$ and $B$ is the set $C$ which consists of all those elements which are present in $A$ but not in $B$ . In symbols, we write,
$A - B = \left\{ {x:x \in A{\rm{ and }}x \notin B} \right\}$
#### Some Properties of Operation of Difference
• $A - B \ne B - A$
• The sets $\left( {A - B} \right)$ , $\left( {A \cap B} \right)$ and $\left( {B - A} \right)$ are mutually disjoint sets.
### Compliment of set
Let $U$ be the universal set and $A$ a subset of $U$. Then the complement of $A$ is the set of all elements of U$U$ which are not the elements of $A$. Symbolically, we write $A'$ to denote the complement of $A$ with respect to $U$. Thus,
$A' = \left\{ {x:x \in U{\rm{ and }}x \notin A} \right\}$ ,obviously $A' = U - A$
#### Some Properties of compliment_of_sets
1. Complement laws:
• $A \cup A' = U$
• $A \cap A' = \phi$
2. De Morgan’s law:
• $\left( {A \cup B} \right)' = A' \cap B'$
• $\left( {A \cap B} \right)' = A' \cap B'$
3. Law of double complementation: $\left( {A'} \right)' = A$
4. Laws of empty set and universal set : $\phi ' = U$ and $U' = \phi$
### Cardinality of the set
• The cardinality of the set defines the number of element in the Set
• If $A$ is the set, Cardinality of the set is defined as $n(A)$
• For $A = \{ 1,2,3\}$ then $n(A) = 3$
#### Set Relations
Joined Set Disjoined Set Set having common elements Set having no common elements $n\left( {X \cap Y} \right) \ne 0$ $n\left( {X \cap Y} \right) = 0$
#### Important Operation on Cardinality
1. If $n\left( {X \cap Y} \right) \ne 0$
$n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right) - n\left( {X \cap Y} \right)$
2. If $n\left( {X \cap Y} \right) = 0$
$n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right)$
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2017-06-28 20:59:42
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https://www.azdictionary.com/definition/beamforming
|
# beamforming definition
• noun:
• a method when the period and amplitude of transmitted indicators are altered, by a feedback procedure, to be able to enhance transmission rate
• A technique where period and amplitude of transmitted indicators are changed, by a feedback procedure, so that you can improve transmission speed
## Related Sources
• Definition for "beamforming"
• a method when the period and amplitude of…
• Sentence for "beamforming"
• Navini, a closely held company in…
• Technology Dictionary for "beamforming"
• Beamforming is some sort of radio…
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2017-11-20 05:56:44
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https://share.cocalc.com/share/5d54f9d642cd3ef1affd88397ab0db616c17e5e0/www/wiki/10(2f)480b(2f)assignments(2f)assn2.html?viewer=share
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Sharedwww / wiki / 10(2f)480b(2f)assignments(2f)assn2.htmlOpen in CoCalc
Author: William A. Stein
10/480b/assignments/assn2
[top] [TitleIndex] [WordIndex]
# Assignment 2, Due 4/19
Do (at least?) five problems from the first section, and both problems from the second section.
### Smaller programs
1. Something nice and easy to get us started: let's write a function fibonacci_list that takes an integer n as input, and returns a list containing the first n Fibonacci numbers.
2. Write a function called flatten, which takes a list and "flattens" it -- that is, removes any internal lists, while keeping the elements in the same order. This is best explained by example:
>>> flatten([1,2,3])
[1, 2, 3]
>>> flatten(["house", [["boat"], 3]])
["house", "boat", 3]
>>> flatten([[1,2], [[[3]], 4, [[5, 6]]]])
[1, 2, 3, 4, 5, 6]
3. Write a function deep_substitute that takes as input a list ls and two non-lists old and new, and walks the list ls, replacing each occurrence of old with new, including in any sublists occurring in ls (or sublists occurring in those sublists, etc). For instance:
>>> ls = [["foo", 3], 5, [[["bar"]]]]
>>> deep_substitute(ls, "foo", 2)
[[2, 3], 5, [[["bar"]]]]
>>> deep_substitute(ls, "spam", "eggs")
[["foo", 3], 5, [[["bar"]]]]
4. Write a function get_name that takes a string of the form
Lastname, Firstname Middlename(s) emailaddress
as input, and returns a valid e-mail alias -- that is, something of the form:
"Firstname Middlename(s) Lastname" <emailaddress>
5. Write a function fixed_point_free_permutations that takes two inputs n and k, and returns a list of k lists, satisfying the following:
• each list is a permutation of the numbers 0 through n-1 (represented as just a list)
• for each permutation, no element is sent to itself (so ls[i] != i for all i and for all permutations ls in the result)
• no entry is mapped to the same thing by two different permutations in the result. (You'll probably find the python library module random useful for this problem.)
Again, here's an example:
>>> fixed_point_free_permutations(5,2)
[[1,2,3,4,0], [2,3,4,0,1]]
whereas this would be invalid:
>>> fixed_point_free_fail(5,2)
[[1,2,3,4,0],[1,3,4,0,2]]
since 0 is mapped to 1 by both of the resulting permutations.
6. Write a function sigma which takes a positive integer n as input and returns the sum of the proper divisors of n (where "proper" just means "don't include n itself). So:
>>> sigma(6)
6
>>> sigma(28)
28
>>> sigma(496)
496
You should be able to do this in one line with a list comprehension -- don't worry about making it fast, just do something naive and easy to implement.
7. Write a function prod which takes a list of integers as input and returns the product of them. You should be able to do this in one line with map/reduce/filter. Using this, implement the "double factorial" -- that is, the function that inputs a positive integer n and returns the product of all the odd integers less than or equal to n.
### Longer Problems
1. It's often nice to print out a readable table of values of a function quickly. Write a function print_table that takes as input a function f and a list ls of inputs, and prints out a nicely formatted table of output values. Here's an example:
>>> print_table(fact, range(3))
+---+--------+
| n | output |
+---+--------+
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
+---+--------+
>>> print_table(fact, [3, 5, 11])
+----+----------+
| n | output |
+----+----------+
| 3 | 6 |
| 5 | 120 |
| 11 | 39916800 |
+----+----------+
In particular, here are the requirements:
• the inputs and outputs should be right-justified, as should the labels n and output
• each side should be wide enough to accomodate all the entries
• there should be at least one space between any entry and the nearest |
• you should make some pretty ASCII art for the border -- it doesn't have to match this exactly, but it should look nice. :)
2. Write a function called copy_and_flatten_directory that takes as input two directory names. It then copies all files in all subfolders of the first directory into the second subdirectory, ignoring the directory structure completely. (That is, it flattens it.) So, for instance, if the first folder had these files
mydir/foo/bar/baz.txt
mydir/foo/bar/pie.pdf
mydir/foo/bar/stuff.txt
mydir/myfile.py
mydir/other_stuff.ss
then, after this function is run, the second folder should contain
baz.txt
myfile.py
other_stuff.ss
pie.pdf
stuff.txt
2013-05-11 18:32
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2019-09-17 00:20:22
|
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https://www.ivdone.top/article/379.html
|
# 数学训练—-数论HDU 2588 – GCD
Description
The greatest common divisor GCD(a,b) of two positive integers a and b,sometimes written (a,b),is the largest divisor common to a and b,For example,(1,2)=1,(12,18)=6.
(a,b) can be easily found by the Euclidean algorithm. Now Carp is considering a little more difficult problem:
Given integers N and M, how many integer X satisfies 1<=X<=N and (X,N)>=M.
Input
The first line of input is an integer T(T<=100) representing the number of test cases. The following T lines each contains two numbers N and M (2<=N<=1000000000, 1<=M<=N), representing a test case.
Output
For each test case,output the answer on a single line.
Sample Input
3
1 1
10 2
10000 72
Sample Output
1
6
260
其中：k>=m
#include <iostream>
using namespace std;
int oula(int n)
{
if(n==1) return 1;
int m=n;
for(int i=2; i*i<=m; i++)
if(m%i==0)
{
n-=n/i;
while(m%i==0)
m/=i;
}
if(m!=1)
{
n-=n/m;
}
return n;
}
int main()
{
int t;
cin>>t;
while(t--)
{
int n,m;
cin>>n>>m;
int ans=0;
for(int i=1;i<=n;i++)
{
if(n%i) continue;
if(i>=m)
ans+=oula(n/i);
}
cout<<ans<<endl;
}
return 0;
}
for(int i=1;i*i<=n;i++)
{
if(n%i) continue;
if(i>=m&&i*i!=n)
ans+=oula(n/i);
if(n/i>=m)
ans+=oula(i);
}
0
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2021-04-11 11:52:49
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33791714906692505, "perplexity": 9232.534864185556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038062492.5/warc/CC-MAIN-20210411115126-20210411145126-00079.warc.gz"}
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http://mbi.dirkjanswagerman.nl/Equity
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Stockholders' or Owners' //''Equity''// is the portion of the balance sheet that represents the capital received from investors in exchange for stock (paid-in capital), donated capital and retained earnings. Stockholders' equity represents the equity stake currently held on the books by a firm's equity investors.
It is calculated either as a firm's
$\large \text{Total Assets} - \text{Total Liabilities}$
or as
$\large \text{Share Capital} + \text{Retained Earnings} - \text{Treasury Shares}$
Stockholders' equity is often referred to as the book value of the company, and it comes from two main sources.
# The first and original source is the money that was //''originally invested''// in the company, along with any additional investments made thereafter.
# The second comes from //''retained earnings''// that the company is able to accumulate over time through its operations. In most cases, especially when dealing with older companies that have been in business for many years, the retained earnings portion is the largest component.
bag
mbi_public
created
Fri, 04 Feb 2011 20:33:15 GMT
creator
dirkjan
modified
Fri, 04 Feb 2011 20:33:15 GMT
modifier
dirkjan
creator
dirkjan
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2017-06-24 06:57:38
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https://www.physicsforums.com/threads/time-for-a-steel-die-to-reach-temperature.844003/
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# Time for a steel die to reach temperature
Tags:
1. Nov 19, 2015
### Thoomy
Hello. I will be doing some experiments with boron steel. The idea is to heat a steel specimen (200x20x1,5 mm dimensions) to 950°C and then quickly transfer it from the oven to the die at room temperature. When the specimen reaches the steel die it will supposedly be at around 800°C. At this temperature, the upper and lower die close and come into contact with the specimen which rapidly cools and quenches in 15 seconds. Since the tool is not cooled, I am now trying to calculate an approximate time when the dies will be at certain temperature (lets say around 60- 80°C) beacuse of the heat conduction from the specimen. The upper and lower die combined are approximately 200x200x150mm in size. Since I will be doing this experiment repeatedly I need at least some kind of orientation how long the experiments will last. Thank you for any help or advice.
2. Nov 19, 2015
### IgorIGP
Assuming that the die material has the same thermal capacity (specific heat) as the specimen can be roughly estimated temperature rise in one experiment:
$$\Delta t= 1+\frac{v}{V}\cdot (t_e-t_r)=1+\frac{200\cdot 20 \cdot 1,5}{200\cdot 200\cdot 150}\cdot (800-20)$$
ie less than one degree if I understand well.
This estimate does not account for the heating associated with stamping. :)
3. Nov 19, 2015
### Thoomy
Well the die material has a bit higher specific heat, but I think the difference is not cruical for the approximate calculation. Correct me if I'm wrong, the result of this equation means, that becuse of cooling down the specimen from 800°C, the die roughly heats up by only 1.78°C in 1 experiment?
4. Nov 19, 2015
### IgorIGP
Forgive me for haste. The formula had to be:
$$t=\frac{t_r\cdot V + t_e\cdot v}{V+v}$$
and as V>>v:
$$t=t_r+ \frac{t_e\cdot v}{V}=20+0,8=20,8$$
so in the next experiment tr=20,8...
That means $$\Delta t = t - t_r = \frac{t_e\cdot v}{V}=0,8$$
:)
5. Nov 19, 2015
### Thoomy
Did not think the difference would be so small :) May I just ask where did you get this equation, since I was searching in a lot of different sources and have never seen so simple solution elsewhere? But thank you for the reply!
6. Nov 19, 2015
### IgorIGP
You have, I sure. You just did not recognize it. The conservation heat energy (balance):
$$c\cdot m\cdot t _m+ c\cdot M\cdot t_M = c\cdot (m+M)\cdot t$$
when two bodies are made of the same material or have equal spesific heats (c). Than assuming the densities are close too we can change the masses by volumes. That is because here it is important the ratio as it is a proportion (the resulting equation for temperature of both bodies t - right side )).
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2018-02-23 21:02:06
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https://cs.stackexchange.com/questions/69273/how-to-write-a-grammar-for-this-language
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# How to write a grammar for this language?
Given the alphabet $Λ = \{a, b, c\}$, i need to write a grammar for this language:
$$L=\{α \mid α∈Λ^+ ∧ aaa \notin α\}$$
I know that in my language the sequence of three consecutive $a$ cannot occur in any string, but how do I make a grammar of it? I'd need the thinking behind the construction of the grammar (like what's the first thing you should consider?), because so far I only manage to go by instinct and this doesn't always work.
• This is a regular language, so you can construct an NFA and convert it into a grammar. – Yuval Filmus Jan 25 '17 at 11:56
• From the NFA I constructed, that's the grammar that came out: $$S → a | b | c | aA | bA | cA$$ $$A →a | b | c | bA | cA | aB$$ $$B → b | c | bA | cA$$ So i should always build the NFA before writing the grammar? With all kind of languages? – emaph Jan 25 '17 at 12:38
• This is just one way to construct a CFG when the language happens to be regular. Of course, there are context-free languages which aren't regular, and this method wouldn't work for them. – Yuval Filmus Jan 25 '17 at 12:40
• Indeed, for example with this non-regular language: $$L= \{ aba^n b^m a^n |n,m>0 \}$$ what's the best way to start? Every string has to start with $ab$, then we have $aba$, and this part has to have the same number of $a$. – emaph Jan 25 '17 at 12:54
• If you have a different question, ask it separately. – Yuval Filmus Jan 25 '17 at 13:02
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2019-09-18 03:19:39
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https://forum.aspose.com/t/is-there-any-way-to-control-conversion-of-image-conversion-on-parse/244219
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We're sorry Aspose doesn't work properply without JavaScript enabled.
# Is there any way to control conversion of image conversion on parse?
Hello, I am trying to import word document with gif file but when I try to create document based on Document(inputStream) after that I have inside document Shape with ImageData which ImageType automatically converted to PNG. As far as understand Aspose converts images automatically (because inside word file I have for sure gif). Is there any was to control this conversion?
@PROCUREMENT2 Could you please attach your input and output document here for testing? We will check the issue and provide you more information.
Unfortunately, there is no way to control this, but Aspose.Words supports GIF images in the model, so it should not be converted to PNG. Though old versions of Aspose.Words really did not support GIF images and converted them to PNG. Which version do you use?
@alexey.noskov I use version com.aspose:aspose-words:22.2
Here is example file
GIF.docx (202.5 KB)
And this file for example code:
Document doc = new Document(
Document doc = new Document("C:\\Temp\\in.docx");
But when you assess ImageData image is converted to PNG. So you get ImageType.PNG. Also, you can notice that ImageType does not have GIF value. This is related to the fact that ImageType returns values that correspond to MSOBLIPTYPE in the Binary Drawing format, which also does not have GIF.
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2022-05-29 04:41:36
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http://mathoverflow.net/feeds/question/73525
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Computing the intersection of dual affine subspaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:46:48Z http://mathoverflow.net/feeds/question/73525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73525/computing-the-intersection-of-dual-affine-subspaces Computing the intersection of dual affine subspaces unknown (google) 2011-08-23T21:22:22Z 2011-08-24T13:01:08Z <p>Suppose we have a convex function , $\phi(x): R^d \to R$. It is well known that the Legendre transform of $\phi$ is also a convex function, and can (loosely) be thought of as the dual or derivative space.</p> <p>Given a hyperplane in the original space of the form $Ax =B$ where $A$ is a $k$ by $d$ matrix in the original space and another hyperplane $C y = D$ in the dual space (where C is a $d-k$ by $d$ matrix), how can I go about computing the unique point determined by the intersection of the two planes? I believe that it is impossible to compute analytically,for example, for the function $\phi(x) = \sum_{i=1}^{d} x_i \log x_i$. However, I'm thinking on finding a good numerical algorithm.</p> <p>There are generic numerical optimization methods - start with a point on one plane, move in the direction of the gradient towards the other - but I've found poor performance trying to do this with the MATLAB toolbox and it also doesn't seem too geometrically insightful. I'm thinking on whether there exists a solution that could somehow exploit the dual nature of the two spaces.</p> <p>Additional comments: Broader comments on the geometry of the situation, or suggestions on how to approach this are also very helpful. Understanding the geometric structure is as important to me as a solution :). Also, if it might ease analysis, consider that the $d$ rows of $A$ and $C$ taken together form an orthogonal basis of $R^d$.</p> <p>Edit: Added some more explanation in the second comment below.</p>
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2013-06-20 01:46:49
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http://mymathforum.com/algebra/36606-equation-3-radicals.html
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My Math Forum Equation with 3 radicals
Algebra Pre-Algebra and Basic Algebra Math Forum
June 16th, 2013, 02:09 PM #1 Newbie Joined: Apr 2013 Posts: 5 Thanks: 0 Equation with 3 radicals Hi I'm having an issue when it comes to solving this equation. ?(x + 2) + ?(x - 3) = ?(3x - 4) I start by squaring, but I don't really know how to multiply 2 radical binomials, ?(x +2) * ?(x - 3) and I can't find any information about it.
June 16th, 2013, 03:08 PM #2 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Equation with 3 radicals You can use the property $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$. You will obviously have to square two times, and then you'll want to make sure no extraneous solutions were introduced, so check ALL of the solutions you obtain in the original equation by substitution.
June 16th, 2013, 03:47 PM #3
Global Moderator
Joined: Dec 2006
Posts: 20,472
Thanks: 2039
Quote:
Originally Posted by Engasaurus I can't find any information about it.
You don't need any information about it. What have you got so far?
June 16th, 2013, 05:06 PM #4
Math Team
Joined: Oct 2011
Posts: 14,307
Thanks: 1023
Quote:
Originally Posted by Engasaurus ?(x + 2) + ?(x - 3) = ?(3x - 4)
a = x + 2, b = x - 3, c = 3x - 4
"squaring twice" means: 4ab = (c - a - b)^2
Thread Tools Display Modes Linear Mode
Similar Threads Thread Thread Starter Forum Replies Last Post Dacu Algebra 16 August 8th, 2013 07:28 AM Dacu Algebra 5 August 8th, 2013 06:49 AM Dacu Algebra 6 May 24th, 2013 01:54 AM Dacu Algebra 21 May 21st, 2013 10:15 PM Mircode Calculus 2 June 7th, 2010 10:39 AM
Contact - Home - Forums - Cryptocurrency Forum - Top
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2019-04-21 18:43:10
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http://scikit-learn.org/dev/modules/generated/sklearn.ensemble.partial_dependence.partial_dependence.html
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# sklearn.ensemble.partial_dependence.partial_dependence¶
sklearn.ensemble.partial_dependence.partial_dependence(gbrt, target_variables, grid=None, X=None, percentiles=(0.05, 0.95), grid_resolution=100)[source]
Partial dependence of target_variables.
Partial dependence plots show the dependence between the joint values of the target_variables and the function represented by the gbrt.
Read more in the User Guide.
Parameters: gbrt : BaseGradientBoosting A fitted gradient boosting model. target_variables : array-like, dtype=int The target features for which the partial dependecy should be computed (size should be smaller than 3 for visual renderings). grid : array-like, shape=(n_points, len(target_variables)) The grid of target_variables values for which the partial dependecy should be evaluated (either grid or X must be specified). X : array-like, shape=(n_samples, n_features) The data on which gbrt was trained. It is used to generate a grid for the target_variables. The grid comprises grid_resolution equally spaced points between the two percentiles. percentiles : (low, high), default=(0.05, 0.95) The lower and upper percentile used create the extreme values for the grid. Only if X is not None. grid_resolution : int, default=100 The number of equally spaced points on the grid. pdp : array, shape=(n_classes, n_points) The partial dependence function evaluated on the grid. For regression and binary classification n_classes==1. axes : seq of ndarray or None The axes with which the grid has been created or None if the grid has been given.
Examples
>>> samples = [[0, 0, 2], [1, 0, 0]]
>>> labels = [0, 1]
>>> from sklearn.ensemble import GradientBoostingClassifier
>>> gb = GradientBoostingClassifier(random_state=0).fit(samples, labels)
>>> kwargs = dict(X=samples, percentiles=(0, 1), grid_resolution=2)
>>> partial_dependence(gb, [0], **kwargs)
(array([[-4.52..., 4.52...]]), [array([ 0., 1.])])
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2018-09-25 20:50:54
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https://www.physicsforums.com/threads/first-order-differential-equation-with-reflected-argument.634147/
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# First Order Differential Equation with Reflected Argument
## Main Question or Discussion Point
I am trying to solve:
(x + 1 + f(-x) )(1 - f ' (x) ) = x+1
f(0) = x_0
x in (-1,1)
I approximated it numerically but any analytic method I try fails. Any ideas?
Related Differential Equations News on Phys.org
haruspex
Homework Helper
Gold Member
Hint: What do you get if you differentiate (x + 1 + f(-x) )? It should make you think of the chain rule.
Ok, so I can make a substitution:
y(x) = x + 1 + f(-x). Then
y ' (x) = 1 - f ' (-x).
I don't see where to go from there, since y ' (x) does not appear in the original equation. However
y ' (-x) = 1 - f ' (x) does appear in the original equation. If i make that substitution I get
y(x) * y ' (-x) = x+1
But I still do now know how to solve that. Is my reasoning correct so far?
haruspex
Homework Helper
Gold Member
Hmm.. yes I was too quick - I thought the minus sign would not be a problem.
However, you can at least try making an assumption about y(-x) and see if it produces a solution consistent with the assumption. I.e., try y(-x) = - y(x), then try y(-x) = y(x).
Judging from the numeric approximation, neither of those seem to be the case. I will try it and see what happens though.
I have tried many things. One thing I was considering was to use the cross correlation of f(x) with a conveniently chosen function, make a substitution, and use the Laplace transform to force the negative sign inside, then hope the inverse Laplace transform works out. That sounds really horribly complicated though.
I have tried a series solution but it did not seem to result in anything worthwhile.
Perhaps a solution does not exist.
haruspex
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2020-05-31 08:25:50
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https://tex.stackexchange.com/questions/180394/figure-placed-on-a-separate-page-is-centered-and-not-on-top
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# Figure placed on a separate page is centered and not on top
I have multiple figures placed in my document (all of which are using [htbp], [ht] etc. and are placed correctly). The last figure receives a separate page (which is alright) but it's centered on this page, which looks weird as the figure is quite small.
Is there any way to get this figure on top of this page? As written above, its a separate page and the last one in this chapter.
I can't really provide a minimal example (since I guess that the position depends on my previous figures), but here's the code I use for the figure (I've tried [ht], [ht!], [htbp] etc but none of them work - I've also deleted the .aux files before compiling):
\begin{figure}[t!]
\centering
\begin{tabularx}{\columnwidth}{cc}
\subfloat[Results 1]
{ \centering \includegraphics[width=0.47\textwidth,height=0.47\textheight,keepaspectratio]{images/plot1}
}
&
\subfloat[Results 2]
{ \centering \includegraphics[width=0.47\textwidth,height=0.47\textheight,keepaspectratio]{images/plot2}
}
\end{tabularx}
\caption{Blabla}
\end{figure}
• Hi and welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. It is the default and in general correct behaviour that floats on pages with no text are vertically centered. Why would you want it another way?
– yo'
May 26, 2014 at 10:33
• It looks a bit weird because the image is quite small. Of course, another option would be to make it bigger (it's a plot), but I'm just wondering if there's a way to place it on top. May 26, 2014 at 10:47
• The usage of ! as a specifier to the figure environment is discouraged for years now ;-)
– user31729
May 26, 2014 at 10:51
• Why are you using [!t] in the first place? How about [!ht] instead if there is room for it here. May 26, 2014 at 11:02
• You could drop the figure environment and use \captionof{figure}{Blabla} instead. Then the graphicx is at the place where you want to have it, if there is enough place left on the page, otherwise on the next page.
– user31729
May 26, 2014 at 11:04
Just place a large enough \vspace after the caption to push it to the top. Specifying the \vspace too large is OK. Here, I place a figure on a [p] page to guarantee the "bad" behavior, but add a 128inch \vspace after the caption, exaggerated to make the point that the actual length is immaterial, if it is large enough to do the trick.
\documentclass{article}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figure}[p]
\centering
\includegraphics[width=0.47\textwidth,height=0.47\textheight]{file}
\caption{Blabla}
\vspace{128in}
\end{figure}
\end{document}
To avoid the "warning" that comes from a too-large float (if that bothers you), one could add negative \vspace prior to the figure contents, as in
\documentclass{article}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figure}[p]
\vspace{-128in}
\centering
\includegraphics[width=0.47\textwidth,height=0.47\textheight]{file}
\caption{Blabla}
\end{figure}
\end{document}
• That's a nasty thing to do that generates a Float to large! warning.
– yo'
May 26, 2014 at 11:56
• @tohecz How does the children's rhyme go?... Sticks and stones can break my bones, but warnings can never hurt me. May 26, 2014 at 11:58
• @tohecz Please see my follow up May 26, 2014 at 12:01
• Warning should make you warned. Warning appearing for things that are correct should be avoided. I always come through the log to check for presence of warning. And these "ok" ones then only confuse me.
– yo'
May 26, 2014 at 12:08
• @tohecz I didn't mean to diminish the import of your statement. You admittedly make a good point, which spurred me to revise my answer. May 26, 2014 at 12:10
As tohecz pointed out, this is the general and correct behavior. But you can change this by using
\makeatletter
\setlength{\@fptop}{0pt}
\makeatother
as found here, following the answer to a similar question.
• Setting lengths globally is not always a good idea ;-)
– user31729
May 26, 2014 at 11:00
• If you deviate from the default, at least do it consistently ;-) @ChristianHupfer May 26, 2014 at 11:32
• @ChristianHupfer I agree with Markus on this. The solution is IMHO the proper way to do it, if you want it in the first place. Upvoted.
– yo'
May 26, 2014 at 11:57
• Thanks a lot! It might sound weird but I like the idea of the centering, it's just the very last page of the paper and looks extremely weird (since a small image is in the middle). May 26, 2014 at 20:14
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2022-10-02 19:28:55
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